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f^
Qf tbe
inniverett/of Midconsin
I
HEAT AND THERMODYNAMICS
IViblUKed b^ the
McGraw-Hill Book. Company
SucccAsors to tKe Rook Departments of the
McGraw Publbhing Company Hill Publishing' Companyr
I\ibljaher5 of books for
Electrical World The Engineering and Mining Journal
Engineering Record American Machinist
Electric Railway «Joumal Coal Age
Metallurgical and Chemical Engineering R>wer
l> WWWWW W WWWW W W W 9 W 9tr W WWWWWW W WW t9WWWWtW9f99WWWW91WWWW9WWWW9WWWWWWWrwmWwii
HEAT
AND
THERMODYNAMICS
BY
F. M. HAETMANN
In dkorff^ of th€ Depcurtmeni of Phynctt and Eledrieal and Meehanieal Eni/inoenng.
Cooper Union Day and Night SchooU
McGRAW-HILL BOOK COMPANY
239 WEST 39TH STREET, NEW YORK
6 BOUVERIE STREET, LONDON, E.C.
1911
COPYBIOHT, 1911
BY
MoGRAW-HILL BOOK COMPANY
163128
TGr
H^5
ERRATA
Page 45, after the equation,
p,«o(l+&T) = po«^(l+ax),
delete remainder of page and read : The foregoing equation follows
immediately from Boyle^s LaWy and by solving for t, we find
^ Pr—po
poa-PrP
Page 239, multiply right hand member of equation (b) by J.
Page 283, first line after equation (28), in place of, '' equation
(27)," read "equation (28)."
HartmAnD's Heat and Thermodynainicw
^gS^giK^gS^gSC)g5HfeS^;SS-^!-
3128 Ca
163128
W\AR
TGr
PREFACE
With so many good works in existence, both on Heat and
Thermodynamics, it may perhaps appear presimiptuous to
publish the following text. The author, however, has long felt
the need of a text, in teaching the subject of thermodynamics,
which properly covers, without introducing too much material,
the fundamental principles of heat measurements. To expect
an average student to cull from his text book on physics, or
some treatise on heat, no matter how well the subject may have
been taught, an introduction to thermodynamics is, in general,
expecting somewhat more of him than he can accomplish. But
it has been found, by experience, that a short cour^ on the
fundamental principles of heat, given as an introduction to the
subject of thermodynamics, greatly reduces the difficulties,
experienced by most students, in pursuing this subject.
Since it is almost impossible for a student to understand a
complex piece of apparatus, imless he can actually examine it,
long and tedious descriptions have been purposely avoided.
Likewise, for the reason that photographs are seldom, if ever,
of any value, all pictorial illustrations are diagrammatic.
It is, of course, impossible to teach the subject of thermo-
dynamics without the application of differential and integral
calculus; but the aim has been throughout to keep within the
bounds of elementary mathematics. However, a fair knowledge
of the calculus, on the part of the reader, has been assumed.
Very few teachers, if any, can present an unbiassed view of a
speculative theory; furthermore, before a student has thor-
ou^ly mastered the groundwork of any subject, he is not in a
position to properly discriminate between the various arguments
that may be advanced, either for or against a speculative theory.
VI PREFACE
It must also be remembered that the average student looks
upon his instructor as an infallible authority; and that he accepts
a theory on the mere say so of his instructor, no matter how
flimsy the arguments upon which it may be based. How fre-
quently one meets those who are in a condition so deplorable
that they can talk very glibly about electrons, ionization, etc.,
and are driven helplessly into a comer by one or two weU
directed questions. Whether there is or is not such a thing
as an atom has nothing to do with the law of definite propor-
tion. Facts will always remain and theories change to fit them.
It is for .these various reasons that speculative discussions, such
as that of the kinetic theory of gases, have been avoided, and that
very hypothetical mediiun — ^the ether — has found no place in
this text. It cannot be too strongly emphasized that before we.
teach metaphysics to a student we must first give him a thor-
ough training in mathematics and physics.
It is the author's opinion that the best that can be done in
any technical course is to thoroughly teach the fundamental
principles underlying the subject, and that it is impossible to
^ve a training which makes the student a practical engineer.
This part must be learned in practice, and the engineer must
keep up to date and in proper touch with his profession by reading
the current engineering literature, and by studying individual
problems as they arise.
It must not be understood that the author expects this text
to supersede such admirable works as Peabody's treatise on
" The Thermodynamics of the Steam Engine," Zeuner's " Tech-
nische Thermodynamik," etc.; but rather as a proper prepara-
tion for the reading of such works. Finally, the author cannot
express his feelings too strongly in regard to the pleasure he
experienced, as a student, while reading Tyndall's "Heat a
Mode of Motion," and Ewing's "The Steam Engjie and other
Heat Engines."
Thanks are hereby expressed to Mr. Albert Goertz for the
care with which he read the manuscript.
F. M. H.
CoopiBB Union, July, 1911.
TABLE OF CONTENTS
HEAT
GHAPTZR PAGB
I. Temperature and Thermal Unitb 1
II. Calorimetrt 11
III. Production op and Eppects op Heat 21
rV. Expansion op Solids and Liquids 34
V. Fundamental Equations op Gases 41
• VI. Elasticities and Thermal Capacities op Gases 68
Vn. Propagation op Heat 80
THERMODYNAMICS
VIII. Fundamental Principles 93
IX. Steam and Steam Engines 116
X. Entropy 128
XI. Applications of Temperature-Entropy Diagrams 137
XII. Elementary Steam and Engine Tests 151
XIII. Compound Engines 180
XrV. Internal Combustion Engines and Fuels 192
XV. Ideal Coefpicient op Conversion and Elemejttary Tests . . . 210
XVI. Compressed Air and Compressors 230
XVII. REPRIGERATIOif 265
XVin. Steam Turbines 289
vu
HEAT
CHAPTER I
TEMPERATURE AND THERMAL UNITS
1. The fundamental conception of hotness or coldness is one
of bodily sensation. That is, an object is said to be hot or cold
depending upon whether it gives us one sensation or another
when we are near it or in contact with-it; and the more intense
the sensation, the hotter or colder the object is said to be. Exper-
ience, however, teaches us that the estimates so formed are not
accurate; since the intensity of the sensation experienced, in
any given case, depends not only upon the condition of the body
under consideration, but very largely upon our experience imme-
diately preceding.
Observation shows that, in general, as bodies are heated or
cooled they change in volume; and in most cases, other things
being equal, bodies increase in volume when heated and decrease
in volume when cooled. Observation fiulher shows that there
is a continual exchange of heat among bodies; i.e., if in any S3rstem
of bodies some are gaining heat, others are losing heat. Or, in
other words, there is a continual tendency toward equilibrium.
2. Temperature. Assimie now, that we are dealing with
two bodies, A and B, and that it is desired to determine which
is the hotter. To do this, some test specimen, upon which
2 HEAT
previous observation has shown a continuous expansion with
continued application of heat, may be used in the following
manner: The test specimen is put into contact with A, and after
a suitable interval of time its length, say, is accurately measured;
it is then put into contact with B, and its length is again measured.
If the length is now greater than it was before, i.e., if the test
specimen expanded when put into contact with S, after having
been in contact with A, B is hotter than A; for, by previous
observation it was found that the test specimen expanded con-
tinually as it became hotter. The body S, therefore, was capable
of imparting more heat to the test specimen than the body A
could impart to it, and B is said to be at a higher temperature
than A. The difference of temperature, then, between two bodies
may be measured by the amount of change in dimensions which
a test specimen imdergoes when, after having been in contact
with one of the bodies, it is brought into contact with the other
body. Such a test specimen is called a thermometer; and, for
accurate measurements, the nature of the thermometer must be
such that no appreciable change is brought about in the body
whose temperature is sought.
If the two bodies, A and B, of the previous discussion, are now
brought into contact, and after a suitable interval of time the
test specimen is put into contact with A and then with B, there
will be no change in its dimensions; i.e., the two bodies are in
thermal equilibrium, or, in other words, at the same temperature*
But, the body B will have lost heat and the body A will have
gained heat; hence, when a body has the capability of imparting
heat to another body, it is said to be at a higher temperature.
Solids, liquids, or gases may be employed in the construction
of thermometers; but it is important that the substance used does
not change its state during the change of temperature. For,
the rate at which a body changes in volume, with respect to
change in temperature, depends upon its physical state; i.e.,
though a substance may exist in the three different states, its rate
TEMPERATURE AND THERMAL UNITS 3
of expansion will, in general, be entirely different in the various
states; being usually the greatest for gases and the least for solids.
The rate of expansion, in general, changes abruptly in passing
from the solid to the liquid, and from the liquid to the gaseous
state.
The three substances most generally employed in the con-
struction of thermometers are: Mercury, alcohol, and dry air] the
most convenient and most commonly used being mercury.
There are two standard temi)eratiu'es, arbitrarily chosen, upon
which all thermometric scales are based; the one is that of melting
ice, and the other* that of the vapor of boiling water under a pres-
sure of one standard atmosphere. The pressure of one atmosphere
being taken equal to that of a column of mercury, at the tempera-
ture of melting ice, whose height is 76 cm., at 45° latitude and
sea level, where the acceleration of gravity is 980.60 cm. per sec.
per sec; or, in c.g.s. units, a pressure of 1.01325X10^ dynes per
square centimeter.
Thebmometric Scales and Thermometers
3. There are Oiree thermometric scales in use: The centigrade
scale, the zero of which is the melting-point of ice, and the point
corresponding to the temperature of the vapor from boiling
water under standard conditions, called the boiling-point, is
marked 100. Hence there are 100 units, called degrees, for the
interval between the melting-point and the boiling-point. The
Fahrenheit scale is marked 32 for the melting-point and 212 for
the boiling-point; hence, there is an interval of 180 degrees
between the two fixed points. The Riaumur scale is marked
for the melting-point and 80 for the boiling-point; hence, there is
an interval of 80 degrees between the two fixed points. Fig. 1
is a diagrammatic representation of the relation of the three
scales. From the foregoing it is obvious that the value of a
degree on the Fahrenheit scale is 5/9 of a degree on the centi-
4 HEAT
grade scale, and the value of a degree on the Reaumur scale is
5/4 of a degree on the centigrade scale. Since both the centi-
grade and Fahrenheit scales are in
common use, it is convenient to
have a simple method of conversion
from one scale to the other.
Let it be desired to convert the
temperature 6 on the centigrade
scale to the Fahrenheit scale. Since
one degree centigrade equals 9/5
degree Fahrenheit, it follows that
an interval of 6*^C. is equal to an
FlG.1. g
interval of — 6®F.; but, since the
5
melting-point on the Fahrenheit scale is marked 32, we must add
32 to give the Fahrenheit reading. Therefore, to convert a
temperature on the centigrade scale to the Fahrenheit scale, we
must multiply the reading by 9/5 and add 32. And similarly, to
convert Fahrenheit to centigrade we must subtract 32 from the
reading and multiply by 5/9.
A method of conversion, which is simpler, is to solve for
that temperature for which both scales read the same. This
temperature is obviously below zero, and is therefore negative.
g
Let 6 be that temperature on the centigrade scale; then, —6+32
5
is the reading on the Fahrenheit scale. But these two readings
are by condition equal. Hence,
from which
6=|e+32;
e=-40.
Therefore, to convert from one scale to the other, the most con-
venient way is to add 40 to the reading, multiply this by the con-
TEMPERATURE AND THERMAL UNITS 5
version factor and deduct 40. To convert from centigrade to
Fahrenheit; add 40 to the reading, multiply by 9/5 and deduct
40. To convert from Fahrenheit to centigrade, add 40, multiply
by 6/9 and deduct 40.
4. Mercurial Thermometer. The mercurial thermometer con-
sists of a capillary tube of, as nearly as obtainable, uniform bore,
on one end of which is blown a bulb. The bulb is filled with
mercury and heated so as to drive out all the air. When this
has been satisfactorily performed, so that nothing but merciu'y
remains in the bulb and tube, the tube is hermetically sealed.
The bulb and tube are then immersed in ice from which the water
is allowed to drain away, and the point to which the mercury
falls marked 'on the stem. Next, the thermometer is immersed
in saturated steam, under standard preasure, and the point to
which the mercury rises marked on the stem. It is, however,
necessary to allow considerable time to elapse between the seal-
ing of the tube and the determination of the fixed points; since
glass, after having been heated, does not, upon being cooled,
immediately return to its original volume. If the bulb contracts
after the fixed points have been placed on the stem, the ther-
mometer will read too high. Joule foimd that the bulb of a certain
thermometer, upon which he had taken observations for twenty
years, was still changing slightly at the end of that time.
After the fixed points are determined, a thread of mercury is
detached, and, by means of it, the tube calibrated to the desired
scale. In this way the units on the scale represent equal voliunes,
and not necessarily equal lengths. But in good thermometers
the tube is of so nearly uniform bore that the lengths of a degree
do not differ by any considerable amount over different parts
of the scale.
Since glass changes in volume when its temperature changes,
it is obvious that the indications of the mercurial thermometer
are proportional to the relative changes between the mercury
and glass, and do not necessarily indicate the true changes in
6 HEAT
temperature; i.e., the indicated temperatures between the two
fixed points depend upon the substances used in construction.
It is readily seen that two mercurial thermometers, if constructed
of different qualities of glass, which have not the same rates of
expansion with respect to mercury throughout the entire scale,
will differ slightly in their readings for some parts of the scale,
even though they read alike for the fixed points.
6. Alcohol Thermometer. The alcohol thermometer is con-
structed in a manner similar to the mercurial thermometer. Its
chief advantage lies in the fact that it may be used for temperatures
below the melting-point of mercury, which is — 39®C. However,
on account of its low boiling-point, which is 78.2®C., alcohol
cannot be employed for high temx)eratures; on the other hand,
the boiling-point of mercury is 357®C.
The discussion of the air thermometer will be deferred imtil
after the discussion of the laws of gases.
6. Thermo Couple. When the junction of two dissimilar
metals is heated, an e,m,f. is developed; and since this e.m.f.
is a fimction of the temperature, such a combination, called a
thermoelectric couple, or simply thermo couple, may be employed
to indicate temperatures. The thermo couple is, in many cases,
where a bulb thermometer cannot be employed, a very conven-
ient device for measuring changes of temperature; and it is par-
ticularly valuable in enabling us to estimate changes of temper-
ature above the boiling-point of mercury. By employing proper
metals, such as platiniun and iridium, very large ranges of tem-
perature can be measured; the meltmg-pomt of iridium bemg
about 2500'*C. and that of platmum about ITTS^'C.
7. Resistance Thermometer. The fact that the ohmic resist-
ance of a metal is a function of its temperature makes it possible
to estimate changes in temperatures, by noting the changes in
resistance of a particular conductor employed for the purpose.
From the foregoing, it is obvious that if any body or combina-
tion of bodies manifest some change, which is a function of the
TEMPERATURE AND THERMAL UNITS 7
temperature and readily measurable, such body or combination
of bodies may be employed to indicate temperatures. But it
is to be carefully noted that the device employed must be such
that the temperature of the body, upon which the measurements
are made, is not appreciably altered by the test body; and that,
in any case, the changes produced in the test body are peculiar
to it, and not necessarily proportional to the changes that would
be produced in some other instrument of a different tjrpe. There-
fore, temperatures must alwajrs be referred to some scale chosen
as a standard. This will be dealt with more fully in the discussion
of thermodynamics.
Heat as a Measurable Quantity
8. If a quantity of water at a temperature ti, be mixed with
an equal quantity of water at a temperature t2, the resulting
temperature of the mixture is very nearly the mean between the
two initial temperatures. If it requires a certain quantity of
heat, g, to raise n grams of water through a given temperature
interval, then it obviously requires a quantity of heat, mg, to raise
mn grams of water through the same temperature interval. If
the water be cooled through the same temperature interval, then
there is imparted to the surrounding bodies a quantity of heat
which is equal to that absorbed by the water while being raised
through that temperature interval.
If the quantities of heat, required to raise a given mass of
water through equal temperature intervals throughout the chosen
thermometric scale, were all equal, then the resulting temperature
obtained when mixing equal masses of water would be the exact
mean between the two initial temperatures. This is shown by
experiment to be very nearly, but not quite true. Hence, there
is not strict proportionality, in the case of water, between change
of temperature, according to our thermometric scale, and change
of heat.
8 HEAT
If equal masses of water and some other substance, say, mercury,
be mbced, the resulting temperature will differ considerably from
the mean between the two initial temperatures. Experiment
shows that if mercury at a temperature ti, be mixed with water
at a temperature T2, the mass of the mercury must be 29.85 times
the mass of the water so as to give a resulting temperature equal
, T1+T2
to-g-.
9. Thermal Capacity. The thermal capacity of a body is
numerically equal to the ratio of change in heat to the corresponding
change in temperature produced by it. The preceding paragraph
states that water has, mass for mass, a greater thermal capacity
than mercury in the ratio of 29.85 : 1; or the thermal capacities
of equal masses of mercury and water are to each other as 0.0335 : 1.
If copper be compared with water the ratio is found to be as
0.0933 : 1. In general, the ratio is less than unity, has different
values for different substances, and varies somewhat with change
of temperature. One notable exception is hydrogen gas, where
the ratio is found to be, at constant pressure, as 3.409 : 1; and
at constant volume, as 2.42 : 1.
To compare different quantities of heat, it is necessary to
choose some substance as a standard. On account of convenience,
water has been so chosen.
10. The Calorie. The quantity of heat required to raise the
temperature of 1 kilogram of water through 1 degree centi-
grade, is called a calorie^ and is the unit adopted for heat measure-
ments. But, since this quantity varies slightly for different
temperatures it becomes necessary, in making exact measure-
ments, to specify some particular quantity. There are three
different definitions for the calorie:
(1) The quantity of heat required to raise the temperature
6f 1 kilogram of water from 0®C. to 1**C., called the zero
calorie.
(2) One hundredth part of the heat required to raise the tem-
TEMPERA.TURE AND THERMAL UNITS 9
perature of 1 kilograin of water from 0°C. to 100°C., called
the mean calorie.
(3) The quantity of heat* required to raise 1 kilogram of
water from 15®C. to 16®C., called the common calorie.
Since it is impossible to realize accurately the first or second
of these, on account of the difficulty experienced in working with
water at 0°C., the last, or common calorie, is the one most gen-
erally used. Furthermore, due to the fact that a great many
heat measurements are made in the range between 15^C. and 25^
C, no large corrections for change in thermal capacity, due to
change in temperature, when the common calorie is employed,
are necessitated; hence, this unit is more convenient than the
others.
Since, in ordinary heat measurements, masses are generally
specified in grams, a secondary unit, called gram calorie, having
the gram instead of the kilogram for the unit mass, is usually
found more convenient than the calorie. In what follows, unless
otherwise specified, by calorie is to be imderstood the quantity
of heat required to raise the temperature of 1 kilogram of water
from 15**C. to 16®C., and by gram calorie, one thousandth part
of the calorie.
11. British Thermal Unit. The thermal unit most commonly
employed in engineering practice, in England and America, is
the British Thermal Unit or B.T.U.; it is the quantity of heat
required, at ordinary temperatures, to raise the temperature of
1 pound of water through 1 degree Fahrenheit.
12. Thennal Capacity per Unit Mass and Specific Heat. The
ratio of the quantity of heat required to raise the temperature
of a given mass of a substance through a given temperature
interval, to the quantity of heat required to raise an equal mass of
water through the same temperature interval, is called the specific
heai of the substance. And, since the unit of heat — ^the gram
calorie — ^is the quantity of heat required to raise the temperature
of 1 gram of water through 1 degree centigrade, it follows
10 HEAT
that the quantity of heat, measured m gram calories, required
to raise the temperature of 1 gram of a substance 1 degree
centigrade, is numerically equal to the specific heat of the sub-
stance; or, in other words, the specific heat of a substance is
numerically equal to its thermal capacity per unit rruiss.
13. Water Equivalent. By the water equivalent of a body
is imderstood the mass of water which has a thermal capacity
equal to that of the given body, and is numerically equal to the
mass of the body multiplied by its thermal capacity per unit
mass.
14. Method of Mixtures. Assume a mass of water mi, at
a temperature xi, to be mixed with a mass m2, of some other sub-
stance, at a higher temperature t2, yielding for the mixture a
resulting temperature of 0. Then, if no heat is lost to or gained
from the surfoundings, during the operation, and the thermal
capacities of the water and substance are sensibly constant for
the temperature ranges experienced, it follows that, since the heat
gained by the water is equal to that lost by the substance, we
must have
mi(0-Ti)=m2c(T2-0); (1)
where c is the thermal capacity per unit mass of the substance,
amd m2C is its water equivalent.
To take a numerical example, assume 268 grams of water at
a temperature 10°C. to be mixed with 1000 grams of mercury at
a temperature 100°C., giving a temperature of 20°C. for the mix-
ture. Substituting in equation (1), we have
268(20 - 10) = 1000c(100 - 20) ;
from which, the thermal capacity of mercury, per unit mass, is
_ 268(20-10 ) _
''" 1000(100-20) "•"'*^^-
CHAPTER II
CALORIMETRY
16. As thermometry has for its object measurement of tem-
peratures, so has calorimetry for its object the measiu'ement of
quantities of heat.
In equation (1), Art. 14, it was shown what must be the
relation between the masses involved, the changes in temperature,
and the thermal capacity per unit mass of a substance, when
two substances are mixed and assume a common temperature.
The actual determination of thermal capacities is, however,
not so simple. Since, in general, the vessel in which the mixing
takes place suffers a change in temperature, its thermal capacity
must be taken into consideration. Furthermore, there is usually
an exchange of heat between the vessel, in which the experiment
is performed, and the surrounding medium during the progress
of the experiment. The vessel, specially designed, in which the
mixing takes place, is called a calorimeter. It therefore follows,
from what has just been said, that in making heat measurements
it is necessary to know not only the thermal capacity of the calo-
rimeter, but also, the rate at which, for a given difference of tem-
perature, exchange of heat takes place between the calorimeter
and the surrounding medium.
16. Law of Cooling. The rate at which a body loses heat to
surroimding bodies ia independent of its thermal capacity, and
depends upon the nature and area of its exposed surface, and the
difference in temperature between it and the surroundings. For
small differences of temperature, i.e., up to a difference of about
11
12 HEAT
15°C., from ordinary room temperature, the rate of cooling is
very nearly proportional to the difference of temperature. This
is known as Newion^s Law of Cooling. The foregoing then states
that the rate at which a body loses heat at any instant is a func-
tion of its surface, and of the diflFerence of temperature between
it and surrounding bodies. Newton's law may be stated as
follows:
f-'^'^ <')
where Q is quantity of heat, t time, K some constant, depending
upon the surface of the body, and t the difference of temperature.
If m is the mass of the body, and c the thermal capacity per unit
mass, then equation (1) may be written
"4'=-^^ ^2)
If c is constant, then equation (2) becomes
dt
<!/ k
where k = — . Separating the variables, we find
mc
from which
and
\o^l = -M;
'kt
T=fcie"^ (3)
To determine the constant of integration fci, assume that we
begin to reckon time when t=ti; i.e., t=ti when < = 0. Making
this substitution, in equation (3), we find A;i=ti; hence, finally
-kt
T=Tie-'^' (4)
CALORIMETRY 13
If m, c, and K are known, thus fixing the value of k, then, by
means of equation (4), the difference of temperature, t, between
the body under consideration and the surroimdings at any time, <,
can be predicted, provided the initial difference of temperature
be known, and the surroundings remain constant in temperature.
It is found, by experiment, that bodies having highly polished
surfaces, other things being equal, lose heat less rapidly than do
those having rough dark surfaces; hence, calorimeters should
have their exposed surfaces highly polished. Furthermore, the
thermal capacities of calorimeters should be small in comparison
with those of the bodies contained in them, and upon which
measurements are being made. Also, while the experiment is
under progress, the calorimeter should be protected from draughts
of air. We are not as yet in a position, nor is it essential, to enu-
merate all precautions that must be taken to give results of abso-
lute precision.
17. Thermal Capacity of a Calorimeter. Since the calo-
rimeter in which the bodies, upon which measurements are to be
made, are contained, always suffers a change in temperature,
it is necessary to know its thermal capacity. But, since the thermal
capacity of the calorimeter, in general, is small in comparison with
that of the bodies upon which measurements are being made,
it follows that a small error in the determination of the thermal
capacity of the calorimeter will not seriously affect the results
obtained for these bodies.
The thermal capacity of a calorimeter may be obtained,
though not with absolute precision, in the following manner:
The calorimeter containing water has its temperature noted,
the water bemg first stirred to msure uniform temperature, then
immediately a quantity of water at some other temperature is
poured into the calorimeter, the contents stirred and the resulting
temperature is noted. If C is the thermal capacity of the calo-
rimeter and stirrer, mi the mass of water originally in it, ti the
temperature of calorimeter and contents before mixing, and
14 HEAT
the common temperature after mixing, then on the assumption
that Ti is higher than 6, the loss in heat, suffered by the calorimeter
and water originally in it, is
(mi+C)(Ti-e).
The gain in heat, by the water poured into the calorimeter, is
W2(0-T2);
where 7712 is the mass, and T2 the temperature of the water poured
into the calorimeter. But, if there are no other heat exchanges,
the loss on the one side must be equal to the gain on the other,
hence
from which
(wi +c)(ti -e) = wi2(e-':2) ;
^__ m2(6--T2)
x:-6 ""' ^^>
There will always be an exchange of heat between the calo-
rimeter and surroimdings; but this can be reduced to a small
quantity by choosing the masses of water such that the result-
ing temperature of the calorimeter is as much below the room
temperature as was its initial temperature above the room tem-
perature. When it is possible, large differences of tempera-
ture, between the calorimeter and room, should be avoided.
18. Cooling Constant of a Calorimeter. When it is impossible
to have the initial and final temperatures differ by equal amounts
from the room temperature — one, of course, being above and the
other below — ^then, to obtain accurate results, correction must
be made, as the case may be, for loss or gain in heat. To do this,
the calorimeter is filled with water, at about 15®C. or 16®C.
above room temperature, to the same height as it will be when
the experiment proper is performed. The temperature b then
CALORIMETRY 15
noted at short intervals of time, the water being continuously
stirred to insure a uniform temperature throughout at any instant.
If the room temperature has remained constant during the
progress of the experiment, then, obviously, equation (4), Art. 16,
applies and the constant k of this equation is determined. In
general, however, better results are obtained by plotting the
observations; using times as abscissas and differences in temper-
atures, between the calorimeter and the room, as ordinates, and
passing a smooth curve through the points so found. The slope
of the tangent then, to the curve at any point, is the rate of change
of temperature at that point; and this slope, divided by the dif-
ference of temperature, or in other words, by the ordinate of the
point, is, according to Newton's law of cooling, a constant for any
point on the curve. Drawing a number of tangents and divid-
ing the slope of each by its ordinate, will give quotients nearly
equal; and the mean of these quotients will be a fair value for
the rate of change of temperature for imit difference of temper-
ature. The rate so obtained multiplied by the thermal capacity
of the calorimeter and contents is numerically equal to the quan-
tity of heat lost, by the calorimeter, per unit time per imit differ-
ence of temperature.
19. Determination of Thermal Capacities. Thermal capacities
may be determined in various ways; the simplest, though not
necessarily the most accurate, and not applicable in all cases, is the
method of mixtures. The substance, whose thermal capacity is
sought, is heated to a temperature ti, which is noted. It is then
inmiediately transferred to a calorimeter of thermal capacity C,
containing a mass of water m2, at a temperature T2. If the result-
ing temperature is 6, and no heat has been lost to or gained from
the surroundings during the operation, then the heat lost by the
one side must be equal to the heat gained by the other; hence we
have
mic(Ti-e) = (m2+C)(e-T2); (6)
16 HEAT
where mi is the mass of the substance, and c its thermal capacity
per miit mass. From equation (6), we find
^, (m.+C)(e-T2)
wi(Ti-e)
which determines the thermal capacity per unit mass of the
substance. If the resulting temperature differs materially from
the room temperature, then corrections will have to be made, from
the curve of cooling of the calorimeter.
20. Method of Cooling. If heat is generated or absorbed
when two substances are mixed, the thermal capacity of a sub-
stance cannot be determined by the method of mixtures. As
an example, if sulphuric acid is mixed with water, considerable
heat is evolved; hence, recourse must be had to some method,
other than the method of mixtures, in determining the thermal
capacity of sulphuric acid. This may conveniently be done by
what is known as the method of cooling.
Assume that we have a calorimeter of known thermal capacity,
C Let the calorimeter be filled to a definite height with water,
and the time noted which is required for the calorimeter and
contents to cool, from a temperature ti to a temperature T2,
when exposed to a definite and constant room temperatiu'e.
Next, the calorimeter is filled to the same height with the liquid,
whose thermal capacity is sought, and the time which is required
to cool from ti to T2, when the calorimeter and contents are sub-
jected to precisely the same conditions as when filled with water,
is again noted. Then, since the average difference of temperature
between calorimeter and contents and the surroundings is the
same in both cases, it follows that the thermal capacities, in the
two cases, are to each other directly as the times required in cooling
through the same temperature intervals. Therefore, if ti is the
time required for the calorimeter and water to cool from ti to
T2, and t2 is the time required for the calorimeter and the sub-
CALORIMETRY 17
stance whose thennal capacity is sought, to cool through the same
temperature interval, it follows that
C+nii __ C+m2C. rgv
^1 fa '
where tni is the mass of water, m2 the mass of the liquid, and c
its thermal capacity per imit mass. From equation (8), since all
quantities excepting c are known, the thermal capacity per unit
mass is determined.
It is, however, not necessary to know the thermal capacity of
the calorimeter. For, if the calorimeter be first cooled through
the temperature interval, ti— T2, when empty, then when filled
with water, and again when filled with the liquid whose thermal
capacity is sought, and the times t, hy and fa are noted, we shall
have
C C+mi C+m2C
t ti fa
(9)
where t is the time required when empty, h when filled with water,
and fa when filled with the liquid whose thermal capacity is sought,
4
and the other symbols having the same significance as before.
Eliminating C, from equation (9), we find
mi(fa— . .
'^m(h^) ^^^^
A convenient form of apparatus, for the method of cooling,
is an alcohol thermometer with its bulb greatly enlarged and
having the form of a hollow cylinder. The substance is then
placed directly inside of the bulb and the whole thermometer,
while cooling, exposed to a constant temperature; this may
readily be brought about by placing the thermometer inside of a
vessel which is surrounded by melting ice.
21. Mechanical Equivalent of Heat From mechanics we
have the following statement: '' The change in kinetic energy
18 HEAT
that a body undergoes in passing over a given path is equal to
the work done in traversing that path." The foregoing statement
is very simple, and readily comprehended when considered in the
purely mechanical sense. That is, it is merely stated that when-
ever a given amount of kinetic energy has been destroyed or
developed in a system, an equivalent amount of work has been
done by the system or, as the case may be, on the system. The
statement, however, does not concern itself with the transforma-
tion of one form of energy into another.
The most general experience, common to all, is that of the
destruction of energy, in the form of mechanical motion, by fric-
tion or impact and the simultaneous evolution of heat. However,
it was not until 1842 that a clearly formulated statement was
made, by J. R. Mayer, to the eflfect, that when heaJt is corwerted into
work, or vice versa, the ratio of the numbers representing the two
quardities involved is constant. Unfortunately, the figures upon
which Mayer based his calculations were in error, and consequently
the value obtained for the mechanical equivalent was also in
error.
Shortly subsequent to Mayer's enunciation. Joule began his
series of experiments to determine the mechanical equivalent
of heat by direct measurement. Joule's method was essentially
as follows: A vessel having fixed vanes, was filled with water,
and a paddle-wheel was caused, by means of falling weights, to
rotate in the water. The fixed vanes prevented the water from
assimiing a rotary motion. The heat developed manifested
itself by a rise in temperature, and the work done was measured
by the weights and distance fallen. A series of experiments was
then made, using mercury instead of water. Another series of
experiments was made by causing one iron plate to rotate with
friction over another iron plate under water.
It is, of course, understood that experiments like these are
attended by great difficulties, and various precautions must be
taken and corrections made which cannot here be enumerated.
CALORIMETRY 19
Still, Joule obtained fairly consistent results; and, the figures he
finally published were not greatly in error.
The results were expressed in meter-kilograms of work per
calorie; i.e., Joule foimd when using water that it required 423.9
m.kg8. of work to develop one calorie, 424.7 m.kg8. of work per
calorie when using mercury, and for the experiment with iron,
the number was found to be 425.2. It must be remembered
that, if comparisons are to be made between results obtained by
experiments, which have been performed in different localities,
corrections will have to be made for variations in the value of g.
Rowland varied Joule's method by using a motor drive instead
of falling weights. The vessel was suspended and the torque
required to prevent rotation measured. Thi3 enabled a much
more rapid expenditure of energy, and a consequent rapid rise
in temperature; thus making the correction due to cooling much
smaller. Rowland's experiment covered the range from 5°C.
to 36**C. Since the thermal capacity of water varies for different
temperatures, a variation was found for the mechanical equivalent
of heat. Rowland found, at Baltimore, where g = 980.0, for the
mechanical equivalent of heat, of the common calorie, 427.3
m.kgs. This may be taken as being substantially correct.
Anthony modified Rowland's method by having a continuous
flow of water through the calorimeter, the temperature of the
inflowing water was constant, and its rate of flow was so regulated
that the vessel was always at room temperature. The mass
of wajer flowing for a given time was determined by weighing,
the number of rotations made by the paddle was recorded on a
speed counter, and the torque was noted, together with the tem-
perature of the inflowing and outflowing water. By this method,
fifince the vessel is always at room temperature, no corrections
for cooling are required; and furthermore, since the vessel suffers
no change in temperature, its thermal capacity need not be known.
The values obtained by this method were in practical concordance
with those foimd by Rowland.
20 HEAT
In the English system of units, the value for the mechanical
equivalent of heat, usually employed, is 778 ft. -lbs. per B.T.U.
Various experiments have been performed, by sending an elec-
tric current through a conductor woimd upon an insulating support
and submerged in water. By noting the current, the applied
6.m./., and the time, during which the current has been flowing,
the energy input is readily computed. Results obtained by this
method agree almost precisely with those obtained by the methods
previously described.
A safe value to use, which if in error is only slightly so, and
which is correct for all practical purposes, is 4.195X10^ ergs per
common gram calorie.
It will now readily be seen that, since we can measure electrical
quantities with great precision, the best method for obtaining a
definite quantity of heat is by sending a steady current for a
given time through a given resistance; it being remembered that
there are alloys whose resistances are practically independent
of temperature.
CHAPTER III
PRODUCTION OF AND EFFECTS OF HEAT
22. Since, in a great many cases, those changes which evolve
heat during their progress, require the application of heat to bring
about a change in the reverse order, it is inadvisable to consider
the production of heat and the effects of heat independently.
Whenever a change of a chemical nature takes place there is
either an absorption or a liberation of heat. In general, heat is
absorbed when a compound is split up into the elements com-
posing it; and heat is liberated when the elements recombine
to form the compoimd. Those compoimds which evolve heat,
during their formation, are called exothermic compounds; and
those rare compoimds which absorb heat, during their formation,
are called endoihermic compounds.
It will now be instructive to consider some particular substance
and the various changes which take place with the continuous
application of heat. Suppose that we are dealing with a definite
mass of ice, under a given pressure, whose temperature is below
that of its melting-point for the applied pressure. (The melting-
point of ice changes slightly with change of pressure; i.e., the
melting-point is lowered about 0.0075°C. for each increment in
pressure equal to one atmosphere.) A definite amoimt of heat
then, must be applied to raise the temperature of the ice to the
melting-point. If heat be then further applied the temperature
will no longer change; but, a change of physical stale takes place
together with a continuous absorption of heat until all the ice is
melted.
21
22 HEAT
23. Heat of Fusion. The quantity of heat required to convert
unit mass of a solid into the liquid state withotU change of tem-
perature is called the heat of fusion of the substance. In the case
of ice, the heat of fusion is approximately 80 gram calories per
gram. If now, after all the ice has been converted into water,
heat be continuously applied, the temperature will rise progress-
ively until, if the liquid be under a pressure equal to one stand-
ard atmosphere, the temperature of 100*^C. is reached. The
temperature will then cease to rise, provided the pressure be
maintained constant, and a change of physical state, namely,
vaporization at constant temperature, takes place progressively
with the continuous application of heat until all the water is
evaporated.
The heat of fusion of ice may be determined as follows: Let
L, m, and ti, respectively, represent the heat of fusion, the mass,
and the initial temperature of the ice; and M, and T2, respectively,
represent the water equivalent of calorimeter and contents, and
the initial temperature of calorimeter. Let the ice now be sub-
merged in the water in the calorimeter imtil it is all melted, and
the calorimeter and total contents assume a common temper-
ature 6. The total heat, then, consumed by the ice in having its
temperature raised from ti to 0, being converted into water at
this temperature, and in raising the temperature of the liquid
from to 0, is
mcTi+m(L+6);
where c is the thermal capacity per unit mass of ice. But this
quantity of heat must be equal to
hence,
From which
ilf(T2-e);
?ncTi+w(L+e) =M (T2-e).
L=-(T2-e)-.(CTi-he).
PRODUCTION OF AND EFFECTS OF HEAT 23
24. Heat of Vaporization. The quantity of heat required
to convert unit mass of a liquid into a vapor mthaui change of
temperature is called the heat of vaporization of the substance.
In the case of water, at lOO^C, the heat of vaporization is approx-
imately 537 gram calories per gram. The heat of vaporization
for most substances becomes less as the temperature rises.
The heat of vaporization of water may be determined by the
method of mixtures as follows: Let superheated steam, at a tem-
perature Ti, be passed into a calorimeter containing water at a
temperature T2. This is continued until a convenient rise of
temperature is obtained in the calorimeter, and a mass of steam
m has been condensed. The quantity of heat, given up by the
steam, is
mc(Ti — T)+wr+w(T— 6) ;
where t is the temperature at which the condensation takes place,
r the heat of vaporization, c the thermal capacity per unit mass
for superheated steam, and the resulting temperature. But
this quantity of heat must be equal to
M(6-T2);
where M is the thermal capacity of the calorimeter and water
initially contained in it. Hence,
mr+m{c(Ti-T)+(T-e)}=M(e-T2);
from which
r=^(0-':2)-{c(Ti-T)+(T-e)}.
To obtain accurate results by means of calorimetric methods,
as previously stated, it is always necessary to take certain pre-
cautions and apply proper corrections.
26. Sublimation. Under certain conditions a substance may
pass directly from the solid to the gaseous state without passing
24
HEAT
through the liquid state. Such a change is called syblimaMon.
Substances such as camphor and iodine, when gently heated,
pass readily from the solid to the gaseous state without melting.
Ice, under normal pressure, also sublimes at temperatures lower
than the melting-point.
26. Superheating. If now, after all of the liquid has been
converted into vapor, heat be further applied, the temperature
will rise progressively with continued application of heat, and the
vapor will become superheated.
All of the foregoing may, instructively, be represented diagram-
matically; bearing in mind that the thermal capacities per unit
mass of water for the three states are: Solid 0.504, liquid 1, and
g(i8e(m8 0.481.
Fig. 2.
Let, as in Fig. 2, the temperatures be taken as ordinates and
quantities of heat as abscissas. Then, if we assume some arbitrary
zero, such as 0, for the initial condition of the ice, we have for the
application of the quantity of heat Oe, the increment of tempera-
ture ea, to the point of fusion. The application of the quantity
of heat ef then brings about the conversion from the solid to the
liquid state at constant temperature. The further application
of the quantity of heat fg brings about th^ elevation of temper-
ature, from the temperature of fusion, to that of vaporization.
The application of the quantity of heat gh brings about complete
PRODUCTION OF AND EFFECTS OF HEAT 25
vapoTziation at constant temperature. The further application
of heat brings about superheating, as shown by the line di.
In general, the physical history of substances with continued
application of heat will be similar to the process just discussed;
but, the ratios of the quantities involved will be entirely dififerent
for each substance.
Assume now, that the process takes place in the reverse order,
step by step; i.e., the steam is first cooled to the point of conden-
sation, then condensation takes place at constant temperature,
and so on, step by step, until the initial condition is reached.
Then during each change, heat is liberated precisely equal in
amoimt to that which was absorbed when the change was taking
place in the opposite direction.
27. Reversible Processes. The changes just described and
depicted in Fig. 2 are, however, not the only changes involved.
Assmne the applied pressure to be maintained constant throughout
the entire change, the volume then of the substance will be chang-
ing continually; and, in general, will be increasing with the tem-
perature. When the volume is increasing, work is being done
by the substance in overcoming the applied pressure. When the
volume is decreasing, work is being done an the substance by the
applied pressure. From the initial condition up to 4°C., this
being the temperature ccM*responding to the maximum density
of water, work is being done on the substance. For all tem-
peratures higher than this, the substance expands continuously
with the continued application of heat; and work is being done
by the substance in overcoming the applied pressure. When the
process takes place in the reverse order, then, wherever heat was
absorbed and work done by the substance during the direct process,
heat will be liberated by, and work will be done on the substance
during the reverse process. And if these quantities be mutually
equal, and no permanent changes have been made to take place
in the surroimding bodies, by this cycle of operations, either
process is said to be a reversible process. That is, when a system
26 HEAT
undergoes a change, or a series of changes, the process is said to be
reversible if, after it has taken place, a second process can be made
to take place, in a manner, such that when the system is again in
its initial condition, there remain, due to these various changes,
no changes oviside of the system.
A little consideration will show that vaporization at constant
pressure, and hence at constant temperature, provided we could
have perfect insidation and no friction, would be a reversible
process. For, under the assiuned conditions, the amount of work
done by the vapor, during its formation, in overcoming the external
pressure, is precisely equal to the amount of work done on the
vapor during its condensation. Furthermore, the quantity of
heat absorbed, during vaporization, from a reservoir of heat at
constant temperature, is precisely equal in amount to the quantity
of heat rejected, to the reservoir, during condensation. Hence,
since all quantities involved balance each other, and no changes
have been brought about in the surroimdings, the process is
reversible.
The cooling and heating of a substance at constant pressure,
together with its consequent changes in volume, can be made
a reversible process only by the aid of a perfect regenerator. The
following discussion will make this clear. Assume that a body
at a temperature Tn, which is also the temperatm-e of the first
reservoir, cools to a temperature ti, by being put, successively,
into contact with n reservoirs, perfectly insulated from each
other, and each reservoir differing in temperature from the one
adjacent to it by an amount equal to (xn— Ti)/(n— 1). The
body then, in cooling, gives up to each reservoir, excepting
the first, a definite quantity of heat, and has done upon it, by the
constant external pressiu-e, a definite amount of work. Let the
process now take place in the reverse order, i.e., the body at a
temperature ti is put into contact with the reservoir at a temper-
ature T2; a definite quantity of heat will be absorbed, which will
be precisely equal to that rejected when put into contact with
PRODUCTION OF AND EFFECTS OF HEAT 27
the reservoir at a temperature ti, after having been in contact
with the reservoir at a temperature t2. Let this be continued
until the temperature Tn is again reached. Now, the work done
by the external pressure on the body, while cooling from the tem-
perature Tn to the temperature ti, is precisely equal in amount
to the work done by the body, in overcoming the external pressm-e,
while being heated from the temperature ti to the temperature
Tn. This process, however, is not perfectly reversible; since the
reservoir at the temperature Tn has given up heat and received
none, and the reservou: at the temperature ti has received heat
and given up none. In the limit, however, as the fraction
(Tn--Ti)/(n— 1), approaches zero for its value, the process becomes
perfectly reversible. But this implies a perfect regenerator;
i.e., a series of reservoirs which are perfectly insulated from each
other, and still have a continuous variation in temperatmre
throughout the series; but this is practically impossible. Hence,
it is obvious that, since in the case of vaporization, we must assume
no radiation and no friction to make the process reversible, and
in the case of cooling and heating of a body, we must assmne a
perfect regenerator and no friction and radiation to make the
process reversible, the process as described and represented
diagranmiatically in Fig. 2 is reversible only in an ideal sense;
i.e., an ideally reversible process,
28. Irreversible Processes. If a constant e.m,f. be applied
to the terminals of a homogeneous conductor, a current will flow
which is directly proportional to the applied e.m.f., and inversely
to the resistance of the conductor. After a time the conductor
will reach a constant temperature; i.e., the rate at which the
energy is being converted into heat by the conductor, due to its
resistance, will be equal to the rate at which heat, expressed in
the same imits, is given to the surroundings by the conductor.
This, however, does not mean that there is thermal equilibriimi.
In this case thermal equilibrium can only be brought about by dis-
connecting the applied e.m./., and consequently, discontinuing the
28 HEAT
dissipation of energy. This being done, the conductor will finally
assume the temperature of the surroimdings, and thermal equilib-
rium will have been established. This process, viz, the con-
version of energy, in the form of an electric cxirrent, into energy,
in the form of heat, differs essentially in one particular feature
from the process discussed in Art 27. The former process,
which we termed an ideally reversible procesSj can be made to take
place, barring various losses, in the reverse order. The latter
process, however, cannot be made to take place in the reverse
order; i.e., it is absolutely impossible to cause a current to flow in
a homogeneous conductor by applying heat to it. Such a process
is called an irreversible process.
Another example of an irreversible process is that of the con-
version of energy, in the form of mechanical motion, into heat by
friction. For it is impossible to restore a system of bodies to
their initial positions by the application of a quantity of heat to
the sm*faces, equal to that which was evolved, due to friction,
during their displacements. The same is true for the case of
impact. When impact takes place between two or more bodies,
a certain amount of kinetic energy is always converted into heat.
But it is absolutely impossible, by the direct application of heat,
to restore the. kinetic energy which was destroyed during impact.
Also, when thermal equilibrium is established by mixing sub-
stances, initially at different temperatures, the process is abso-
lutely irreversible.
29. Dissociation. Under Art. 26 we discussed the physical
history of water with the continued application of heat up to the
point of its superheated vapor. If heat be still further applied
to the superheated vapor its temperature will continue to rise,
up to some very high temperature, when complete dissociation
takes place; i.e., the vapor splits up into its two constituent
elements, viz, hydrogen and oxygen. Such a change is called a
chemical change. If, while the pressure is maintained constant,
heat be further applied, the gaseous mixture will rise in temper-
PRODUCTION OF AND EFFECTS OF HEAT 29
atiire and increase in volume progressively with continued appli-
cation of heat. If now, the process be reversed, i.e., the mixture
be cooled, the temperature and volume will diminish until the
temperature of dissociation is reached. When this point is reached
the two gases recombine to form steam, and precisely the same
amount of heat is evolved as was absorbed to bring about decom-
position. The quantities of heat, however, which are involved
in chemical decomposition and recomposition are very large in
comparision with those quantities involved during changes in
temperature and changes of physical state. Indeed, our greatest
source of supply of energy, in the form of heat, is that due to
chemical combination; viz, the combination of carbon, in the form
of coal, with oxygen.
It is true that the amount of dissociation is a function of the
temperature; i.e., even water at ordinary temperatures has a
small percentage of dissociation. This, however, does not
invalidate the statement that, the energy absorbed during dis-
sociation is equal to that liberated upon recombination.
To give an illustration of the quantities of heat evolved during
chemical combination, we may take as examples the combination
of hydrogen and oxygen to form steam, and the combination of
carbon and oxygen to form carbon-dioxide. In the former case,
1 gram of hydrogen combining with oxygen to form steam
(H2O), about 34,000 gram calories are evolved, or expressed in
mechanical units, 1.43X10^^ ergs. In the latter case, i.e., when
1 gram of carbon combines with oxygen to form carbon-dioxide
(CO2), about 8000 gram calories are evolved, or, expressed in
mechanical units, 3.36X10^^ ergs.
'30. Electrolysis. Dissociation may also, in general, be brought
about by electrolysia. That is, if an electric current be passed
through a chemical compound, in the form of a solution, the com-
poimd will be split up into its constituents. '
Suppose that we are dealing with a solution of copper sulphate
(CUSO4), and that the two electrodes are absolutely inert as regards
30 HEAT
chemical reactions. Then if an electric current be passed through
the solution^ copper will be deposited on the negative electrode
{cathode), and the radical SO4 will be liberated at the positive
electrode {anode). The SO4 thus liberated will combine with
hydrogen of the solvent to form sulphuric acid (H2SO4) ; and at
the same time oxygen will be liberated. The equation, repre-
senting this reaction, is
CUSO4+H2O = H2SO4+CU+O.
K an electric current be passed through water, then the water
will be split up into its two elements, hydrogen and oxygen;
hydrogen being given off at the cathode and oxygen at the anode.
In the case of dissociation by heat, a definite quantity of heat
disappears for a given amount of dissociation; and an evolution
of an equal quantity of heat upon recombination. But, since
a given quantity of heat represents a definite amount of energy,
it follows that dissociation involves storing of energy. Likewise,
when dissociation is brought about by electrolysis a definite
amount of energy is consumed for a given amoimt of dissociation,
which must necessarily be equal to that consumed when the same
amount of dissociation is brought about by the application of
heat; since the energy stored is the same in amount for both
cases. It is true that a solution becomes heated when conveying
a current; but, this has nothing to do with the dissociation. The
development of heat being merely due to the resistance of the
solution, the same as when any other conductor is conveying a
current.
It is immaterial, so far as the foregoing argument is concerned,
whether we consider the solution initially partly ionized and the
current merely a icarrier of the ions, or that the current actually
splits up the compound.
31. Faraday's Discoveries. Faraday showed experimentally
that the amoimt of dissociation is directly proportional to the
PRODUCTION OF AND EFFECTS OF HEAT 31
time and the intensity of the current; and furthermore, that the
amoimt of chemical action is the same for all parts of the circuit.
The latter part may perhaps be best illustrated as follows : Assmne
that there are two voltameters connected in series; the first
containing a solution of copper-sulphate and the second water.
Then upon passing a steady current through the circuit a definite
amount of copper will be deposited on the cathode of the first
voltameter, for a given interval of time, and a definite amount of
hydrogen liberated at the cathode of the second voltameter,
during the same interval of time; and these two quantities mil be
in the same raiio as their chemical combining numbers. That is,
for every gram of hydrogen set free at the cathode of the second
voltameter, 31.59 grams of copper will be deposited on the
cathode of the first voltameter; where, if hydrogen be taken as
unity, 31.59 is the chemical equivalent of copper in copper-sul-
phate. During the same time that the 31.59 grams of copper
are being deposited on the cathode of the first voltameter, 1
gram of hydrogen must be liberated to combine with the sulphion
(SO4), set free to form H2SO4.
32. Counter Electromotive Force. Since the amount of dis-
sociation, other things being equal, varies directly as the cur-
rent, and the amount of energy stored during dissociation depends
upon the compound dissociated, it follows that every compound
offers a definite counter e.m.f, to dissociation. And any applied
e.m.f. less than this cannot bring about dissociation. To make
this clear, the e.m./. necessary to dissociate water will here be cal-
culated. The amount of hydrogen set free, per coulomb of elec-
tricity conveyed, is 0.000010357 grams. The number 0.000010357
is called the electro chemical equivalent of hydrogen.
Let, in the cg.s. system of units, z be the electro chemical
equivalent of hydrogen, I the current, and t the time, then the
mass of hydrogen liberated, during the time t, is
m^Izt (1)
32 HEAT
Let h be the heat, expressed in mechanical units, required to
dissociate 1 gram of hydrogen, then
mh=IEt; (2)
where E is the applied e.m./. Substituting in equation (2), the
value of w as given in equation (1), we find
Izth-^IEt;
from which
zh^E (3)
If now, in equation (3), we substitute for z and h their values,
remembering that for 1 gram of hydrogen, combining with oxygen
to form steam, A = 1.43 X 10^^ ergs; and that z, in the cg.s, system
of units, equals 0.00010357 grams per imit quantity of electricity,
we find
or,
£?= 1.43X1012x0.00010357 c.g.8. units e.m/.,
„ 1.43X10^2x0.00010357 , ^^ u
E = r>.^ = 1.48 volts.
10**
It must, however, be emph&sized that, in general, the e.m/.
for most cells is a function of the temperature; and, therefore,
to calculate the e.m.f, this function must be known. We are not
prepared, here, to take up this matter. The coimter e.m.f. is
readily determined by experiment.
33. Junction of Dissimilar Metals. If the junction of two
dissimilar metals be heated an e.m.f. is developed which is a func-
tion of the temperature, and the metals which form the junction.
To take a concrete example, assimie a junction of antimony and
bismvih. Such a junction, if heat be applied to it, develops an
e.m.f. which tends to send a current from bismvih to antimony;
and if a current be sent from bismuth to antimony, by the applica-
PRODUCTION OF AND EFFECTS OF HEAT 33
tion of an external e.m.f.y there will be a tendency to reduce the
temperature of the junction. On the other hand, if a current be
sent through the junction from' antimony to biamvih, heat will
be developed.
31. As a r^mn^, we may then state that the most general
effects of heat are: To change the volumes of bodies; to bring
about physical changes of state; in general, to promote chemical
dissociation; to develop an e,m,f. at the junction of dissimilar
metals.
For the prodiiction of heat we may state the following examples:
The production of heat by the mechanical compression of bodies
which expand upon the application of heat; when the physical
state of a body changes in the reverse order from the change when
heat is applied; in general, by chemical combination; at the junc-
tion of two dissimilar metals when a current is passed in a direc-
tion opposite to that of the developed e.m.f. when heat is applied.
Also, the production of heat when an electric current is conveyed
by a homogeneous conductor; and, in general, when friction is
being overcome, and mechanical motion destroyed by impact.
36. The Principle of Energy. The various relations discussed
in this chapter may now be summed up and stated quantitatively
in a very simple manner. This generalization, known as the
principle of energy, or conservation of energy, is one of the most
extensive of generalizations, and may be stated in substance as
follows: If in any system, from which no energy escapes and into
which no energy enters, account be taken of aU forms of energy,
then no maUer what transformations take place within the system,
the sum total is a constant quantity. So far as experience goes,
the foregoing statement is consistent with all phenomena; and
hence, in all subsequent demonstrations its truth will be assiuned.
CHAPTER IV
EXPANSION OF SOLIDS AND LIQUIDS
36. Linear Expansion. It has been previously stated that,
in general, bodies expand when the temperature is augmented.
It is found by experiment that a body, such as a metal rod,
increases in length by approximately equal amounts, between
0°C. and lOO^C, for equal increments of temperature. But,
even though there is an approximate proportionality between
change in length and change in temperature for moderate ranges,
such as just specified, it must not be inferred that this is generally
true for large ranges or high temperatures.
If a body of unit length expand in length by an amoimt a
for unit increment of temperatiu"e, then a body of length I will
expand in length by an amount Za for an increment of 1 degree;
and for an increment of t**, between 0°C. and lOO^C, the body
will expand in length, approximately, by an amount Zocr. Hence,
if the length of the body at 0** be denoted by lo, and at t** by Z„
we have
from which
^ = fo(l+flpr) (1)
The quantity a is called the coefficient of linear expansion, and
may be defined as the ratio of the change in length, per unit
change in temperature, to the length at zero. The quantity in
the parenthesis, viz, l+or, is called the /ocfor of linear expansion.
34
EXPANSION OF SOLIDS AND LIQUIDS 35
37. Voltuninal Expansion. Homogeneous isotropic bodies
will change in amount by like fractional parts of their original
dimensions in all directions when the temperature changes.
Assume that we are dealing with a rectangular parallelopiped
whose three edges at zero temperature are oo, bo, and co, its volume
then, at zero, is
»o=ao&oco. . (2)
If the temperature, now, be changed to t®, the three edges become:
ao(l+ore), 6o(l+ore), and co(1+oct); from which we have, for the
volume at t°,
^=ao6oco(l+opc)3 (3)
Substituting in equation (3), for ooioco the value vo, as given by
equation (2), we have
t;,=«o(l+(rr)3 (4)
Expanding equation (4), we find
v,=vo(l+S(rz+3a?z^+Q?'^) (5)
Now, a is a very small quantity in ^comparison with the dimen-
sions of most bodies; hence, the two terms containing a^ and a^
may be neglected; and we have
v,=vo(l+Sopz) (6)
Equation (6) is, of course, only approximately true; but, that it is
correct for most practical purposes will be evident from an
inspection of 'the following table, which gives the coefficients of
linear expansion, per degree centigrade, for a few solids. It must
be remembered that for substances like brass and glass the nmn-
bers are only approximate; and in any case the value found for
the coefficient of expansion will depend somewhat upon the treat-
ment to which the specimen was subjected in its preparation.
36 HEAT
Coefficients of Lineab Expansion
Platinum CSSOXIO"*^
Copper 1.678X10"*
Steel (annealed) 1.095X10"*
Zinc 2.918X10"*
Brass 0.187X10"*
Glass 0.083X10"*
Invar (steel containing 36% nickel) 0.087X10"*
It is only necessary to substitute the values of a, as given in the
foregoing table, in equation (5) and it becomes evident that
equation (6) is approximately true. Hence, we may say that-, the
coefficient of voluminal expansion is practically equal to three
times the coefficient of linear expansion.
38. Non-Isotropic Bodies. There are certain bodies which
have different physical properties in diflferent directions. Such
bodies are termed non48otropic. A notable example is that of
Iceland spar, in which it is found that the coefficient of linear
expansion in one direction is 2.63X10"*; whereas, in a direction
normal to this it is found to be only 0.544 X 10"*.
It is also interesting to note that Iceland spar manifests
different optical properties in different direltions,
39. Expansion of Liquids. Liquids, in general, change more
. rapidly in volume than do solids for equal changes in temperature.
However, since, in the case of liquids, the term linear expansion
is meaningless, we deal only with voluminal expansion.
The determination of the coefficient of linear expansion is
quite simple; hence, the description of the methods employed in
its determination was omitted. The determination of the coef-
Jtdent of voluminal expansion of a liquid is attended by various
difficulties; and, since the discussion of the principles involved will
proves instructive, a few methods will be described.
EXPANSION OF SOLIDS AND LIQUIDS 37
Assume that we have a glass flask tennmating in a tube of
capillary bore, and that the mass of the flask when empty is known.
The flask is then filled, to a definite mark on the capillary tube,
with the liquid at a temperature ti, whose coefficient of voluminal
expansion is sought. The mass of the vessel and contents is now
determined. The difference between this mass and the mass of
the flask gives us the mass of the liquid at the temperature ti.
If Di is the density of the liquid at the temperature ti, and V
the volmne of the liquid in the flask at the same temperature,
then
J5i=-pr; (7)
where Mi is the mass of the liquid in the flask at the temperature
Ti. Let, now, M2 be the mass of the liquid in the flask, when its
temperature is T2, the flask being filled to precisely the same
mark on the tube as when the temperature was ti. The volume
of the flask will now be
r = F{l + P(T2-Ti)};
where ^. is the coefficient of voluminal expansion of tfie glass of
which the flask is composed. ^ may be computed from the coef-
ficient of linear expansion, or determined xlirectly by experiment,
as will be shown subsequently. The density of the liquid at T2,
will now be
^^-l^~Fil + &(T2-.l)} ^^^
Dividing equation (7) by equation (8), we find
38 HEAT
If the coefficient of voluminal expansion of the liquid, between
0° and the other two temperatures under consideration, is approx-
imately constant, we have
where Do is the density of the liquid at 0** and a the coefficient of
voluminal expansion. Similarly, we find
Z)2=j^; (11)
l+aT2
from which, by dividing equation (10) by equation (11), we find
Di l+aT2
D2 1+aTi
(12)
Finally, by equating the right-hand members of equations (12)
and (9), we find
•±S,^.|,+M.-,.)l (.3,
In equation (13), all quantities excepting a are known; hence,
its value is determinate.
Another method is as follows: A solid, whose coefficient of
voluminal expansion is accurately known, and which does not react
chemically with the liquid whose coefficient of voluminal expansion
is sought, is weighed in the liquid, first at temperature ti, and
second at temperature T2. The weight of the solid being known,
its loss of weight for the two temperatures is known; and, smce
the volume of the solid for any temperature may, from its known
coefficient of voluminal expansion, be computed, the densities
of the liquid for the two temperatures ti and t2 are readily found,
and from these, as previously shown, the coefficient of voluminal
expansion.
EXPANSION OF SOLIDS AND LIQUIDS
39
■10 'a^
?
i-_
40. Direct Measurement of CoeflSicient of Voluminal Expan-
sion. The method about to be described, and the one by which
Regnault determined the coefficient of expansion of mercm-y,
depends upon the principle that when commimicating columns
of liquids are in equilibrium their heights are inversely as their
densities. In Fig. 3, AB and CD are two vertical iron tubes,
cross connected by the horizontal tube BD. The two horizontal
tubes, AE and CF, terminate in the vertical tubes EO and FI,
which are connected by the inverted glass U-tube GJI, The
tube containing the stop cock s,
is connected to the receiver of a
compression pump; and after the
apparatus has been filled with
mercury, air is forced in from
the compressor until the mercury
in the tubes EG and FI is at
a convenient height. The stop
cock B is then closed. If, now, the
temperature, and consequently
the density, of the mercury in the
tubes AB and CD be the same,
the columns in EG and FI will be
at the same level. On the other
hand, if the temperatures of the two columns AB and CD be not
the same, then the colmnns in EG and FI will not be at the same
level. For, since there is free communication between B and
Z), the pressure must be the same at these two points. Further-
more, the pressure of the air inside the U-tube being everjrwhere the
same, and, since this pressure plus the pressure, due to the column
in FI, balances the pressure due to the column DC, and the same
pressure plus the pressure, due to the column in EG, balances that
of the column BA, it follows that if the pressures, due to the two
columns AB and CD, are not the same, the two columns in EG and
FI cannot be at the same level.
■I'
■d
I
E
Fig. 3.
1
40 HEAT
Suppose, now, that the column in EO, FI, and CD be main-
tained at 0°C. and the column AB at t°C., th«i the column in FI
must exceed the column in EG jin height, by an amount A, such that
(i?-A)(l+orr)=ff; (14)
where a is the coefficient of voluminal expansion of mercury.
From equation (14) we find
41. If, now, it be desired to determine the coefficient of expan-
sion of some other liquid, it becomes only necessary to take a
flask and determine its mass when empty, then the mass of flask
and contents when filled to a given mark at various temperatiu*es,
first with the liquid whose coefficient is sought, and then when
filled with mercury to the same mark for the various temperatures.
From the mass of mercury required to fill the flask at various
temperatures, and from the known density of mercury for these
temperatures, the volume of the flask is readily computed. From
the known volume of the flask and the mass of liquid required to
fill it at the various temperatures, the densities corresponding
to those temperatures are found.
CHAPTER V
•FUNDAMENTAL EQUATIONS OF GASES
. 42. Isofhermal Equation. Experiment shows that, between
certain limits, for the so-called permanent gases, such as hydrogen,
oxygen, nitrogen, etc., the product of pressure and volume is
a constant, for constant temperature. Expressed symbolically
?w=*; (1)
where p is the applied pressiu^ per unit area,* v the corresponding
volume of the gas, and k some constant whose value depends
upon the units chosen. Equation (1) is usually designated as
Boyle's Law. But, in any case, the equation which expresses the
relation between the pressure and volume of a gas, at constant
temperature, is its isothermal equation.
43. Gay-Lussac's Law. As a fiuther result of experiment
it is foimd that all gases which obey Boyle's Law have the same
constant temperaiure coefficient; i.e., all gases, under constant
pressure, expand by the same fractional part of their voliunes at
zero temperature, for equal increments of temperature. This
is known as Oay-Lussac's Law. If we denote by a the increment
in volume, for a imit volmne of a gas, under a constant pressure
PO; when its temperature changes from zero to imity, then the
volume of a gas at t degrees is
v^=vo+VQ(xz) (2)
* In all subsequent equations, unless otherwise stated, p will be used
to denote pressure per unit area.
41
42 HEAT
where «o is the volume of the gas at zero temperature, and v^
the volume, under the same pressure, at t degrees. Writing
equation (2) in another form, we have
t;, = t;o(l+opr) (3)
If, after the temperature t, and the corresponding volume
Vtf under the constant pressure po; has been attained, the pressure
is augmented, the temperature being maintained constant, until
the gas assumes its original volume vo, we must, from Boyle's
law, have the following relation:
p,=po(l+oeT) •. . . (4)
As a matter of fact the experiment is most conveniently per-
formed, by varying the pressure, so as to maintain the volume
constant, as the temperature is varied. Equation (4) may there-
fore be considered as being the expression of experimental results.
Multiplying both sides of equation (4), by vo, we obtain
Pt«^o=Pot^(l+aT) (6)
If, now, while the temperature is maintained constant, the pressure
be varied, the volume will vary according to BoyWs Law; i.e.,
pt;=p,vo=po«o(l+aT); (6)
where p is any pressure and v the corresponding volume at the
temperature t.
Since Gay-Lussac^s Law also holds for temperatures below
zero, a, of course, being a decrement, we have
pv=povo{l — (xx) (7)
Equation (7) reduces to zero when ore equals unity; hence
T = l/a is the temperature below zero for which pv=0.
FUNDAMENTAL EQUATIONS OF GASES 43
For the centigrade scale a =0.003665, very nearly; hence,
T = 1/0.003665 = 273, very nearly. Therefore, if there were no
deviation from the relation expressed by equation (7), then at
— 273®C. the product of pressure and volume would become
zero. As a matter of fact, all gases liquefy at temperatures above
-273°C. This point, 273°C. below zero, is called the ''absolute
zero "; and temperatures measured from this zero are called tem-
peratures on the " absolute scale,"
Substituting now, in equation (6), for a its value, viz, 1/273,
we have
?w=Pot;o(l+273)=|^'(273+T). ... (8)
But, in equation (8), pofo/273 is a constant for any particular mass
of a gas, and 273 +t is the temperature as measured on the
" absolute scale" Replacing the former by R and the latter by
r, equation (8) becomes
VV=RT (9)
The numerical value of R depends, of course, upon the units
chosen.
Equation (9) is called the characteristic equation of a gas, and
shows that the product of pressure and volume is directly propor-
tional to the temperature as measured on the " absolute scale." *
44. Departure from Boyle's Law. For ordinary pressures,^
Boyle's law is approximately true; i.e., for air the ratio of the »
product of pressure and volmne, when the pressure is one atmos-
phere, to the product of pressure and volume, when the pressure
is two atmospheres, is about 1.002. The ratio of the initial value
of p« to the final value of pv becomes greater as the final pressure
becomes greater, imtil certain very high pressures, which are dif-
ferent for the various gases, are reached; after which the value
* Since aU knowledge is relative, the expressions ''absolute zero^' and
''absolute scale" are not well chosen.
44 HEAT
of pv increases rapidly with increase of pressure. The values of
the pressures for which, at ordinary temperature, the product
pv is a minimum, are as follows : 100 meters of mercury for oxygen,
65 meters for air, and 50 meters for both nitrogen and carbon-
dioxide. For hydrogen the deviations from Boyle's Law are less
and in the opposite direction.
Since Boyle's Law is not rigidly true, it follows that all equations,
which have been based on it, are not rigidly true. However,
for ordinary ranges of pressure and temperature the character-
istic equaiion is very nearly true. And, for the mathematical
discussion of gases, it is very convenient to assume that we are
dealing with a gas for which the characteristic equation is rigidly
true. Such a gas is called a perfect or ideal gas.
46. Gas Thermometer.* If a gas be confined in such a
manner that its volume is maintained constant while the tem-
perature varies, then it follows, from equation (9), that between
those limits for which the equation is approximately true, the
pressure varies directly as the temperature. All that is necessary
then, to enable us to measure temperatures accurately, is a glass
bulb of convenient size, filled with a gas, the most convenient being
dry air, and some device by means of which the pressure can be
regulated and measured. Such an arrangement constitutes a
gas thermometer. Fig. 4 is a diagrammatic representation of
the arrangement. A is a glass bulb, filled with gas, and connected
by an inverted capillary U-tube, to the U-tube (Jbc. S is a vessel,
open at the top, partially filled with mercury, and communi-
cates, by means of a flexible tubing, with the U-tube at 6. The
tube 6c being open at the top, the mercury in B and he must be
at the same level. Assume that, when the temperature of A is
at zero, the mercury in S, in 6c, and in 6a is at the same level;
then, if the temperatiu-e of the bulb A, and consequently that of
* In all subsequent demonstrations, unless otherwise specified, the
symbol T will be employed to designate temperatures as measured by
the ideal gaa thennameter, the zero of which is about — 273°C.
FUNDAMENTAL EQUATIONS OF GASES
45
o
the gas in it, be increased, the gas will expand if the pressure be
constant. Hence, to maintain the gas at its original volume,
the vessel B must be raised so as to increase the column be, in a
manner, such that the column ba, and hence the volume of the gas.
is maintained constant. The
difference in the height of the
colmnns be and ba measures
the increment in pressure; and
hence, the increment in tem-
perature may be calculated. A
correction must, however, be
made for the change in volume,
due to change in temperature,
for the bulb A. This correction
is readily applied, provided the
coefBcient of voluminal expan-
sion for the glass, confining the
gas, be known.
Let po and ro, respectively, rep-
resent the pressure and volume
of the gas for the temperature at zero, p, the observed pressure
for the temperature t, a the coefficient of expansion for the gas,
and P the coefficient of voluminal expansion for the glass. The
new volimie, then, for the gas at the temperature t, will be wo(l + ^t) ;
hence
Px«o(l + &t) = pot^(l + ax) .
Fig. 4.
Now, pT==Po(l+y^); where a' is the /Apparent
expansion of the gas. Making this ysubstitutio)
eliminating, w/ find
(l-Ha'T)(l + M/l+aT;
from whicl
/
ifficienlf oj
for p/ axu
J'etf Eifratt
"doyJe-'J Law, miJ :s<>lvinj ;f'>y'x', "^^-find
T- f*r-p
/v*-/v^*
46 HEAT
When extreme precision is sought, corrections must also be
made for changes in volume of the bulb, due to changes in pressure.
Gas thermometers are used only for purposes of standardization.
46. Expansion without Doing External Work. Experiment
has shown that there is no energy consumed in the simple expan-
sion of a gas; i.e., when a gas expands in such a manner that no
external pressure is overcome, and hence, no external work is
being done, no energy is consumed by it. This experiment was
first performed by Gay-Lussac (who apparently did not realize
its full significance) in the following manner: Two vessels, one
of which was exhausted and the other filled with air under a pres-
sure, were placed in a calorimeter and surrounded by water.
When the stop cock in the tube, connecting the two vessels, was
opened, the air from the one vessel expanded into the other,
bringing about an equalization of pressures. During this process
the gas increased in volume without doing any work external
to the system. That is, work was done by the gas under a high
pressure, in the one vessel, in expanding against the increasing
pressure of the gas in the other vessel. But, since the temperature
of the system after the completion of the process was found to
be precisely equal to that of the system before the process *began,
it follows that the total energy of the system is unchanged by the
expansion. If energy were required to bring about an increase
in volume, the temperature of the system at the end of the process
would necessarily be less than at the beginning of the process.
Or, to express the result in another way, since the temperature of
the system is unchanged by the change in volume, the work done
by the gas in the vessel, initially under the higher pressure, is pre-
cisely equal in amount to the work done on the gas in the other
vessel. The results just deduced may be embodied in a simple
statement; i.e., the intrinsic energy of a perfect gas is a function
* only of the temperature. Or, to put it still in another way, the
heat of disgregation of a perfect gas, when no external work is
being done, is zero.
FUNDAMENTAL EQUATIONS OF GASES 47
The foregoing experiment was subsequently repeated, in the
most careful manner, by Joule and found to be approximately,
though not rigidly, true.
47. Thermal Capacities of Gases. If a gas, under a pressure
p, expand by an amount in volume dv, the external work done will
be numerically equal to pdv; where p is the pressure per unit
area. This is shown as follows: From the definition of work,
we have
dw-^Fds] (10)
where F is the applied force and ds the displacement. But, since
Fy the applied force, is nimierically equal to the product of p,
the pressure per imit area, and the area A, we have
dw=pAd8 (11)
But
Ad8=dv)
hence, by substituting in equation (11), we have
dw=pdv (12)
Since, according to the experiments of Gay-Lussac and Joule,
when the temperature of a gas is augmented, that part of the heat
which is required to elevate the temperature of the gas is practically
the same for equal ranges of temperature, no matter whether the
volume be varied or maintained constant, it follows that the quan-
tity of heat required to bring about a given elevation of temper-
ature, when the pressure is maintained constant, is greater than
the quantity of heat required to bring about an equal elevation
of temperature, when the volume is maintained constant. For,
in the former case, heat is required, not only to elevate the temper-
ature of the gas, but also to do the external work due to the expan-
sion of the gas; whereas, in the latter case, the heat consimied is
only that required to elevate the temperature of the gas.
48 HEAT
The ratio of change in heat to the corresponding change in
temperature, in a unit mass of gas, when the volume is maintained
constant, is a measure of the thermal capacity per unit mass at
constant volume, and is denoted by Ct. Hence, for a unit mass
of gas, we have
(§).--'
(13)
In a like manner
(14)
where C, represents, for a gas, the thermal capacity per unit nuiss,
under a constant pressure.
The ratio Cp/Cv=n is practically a constant for all the per-
manent gases; and, in the case of air, is approximately 1.405. The
quantity (Cp—Cv), is evidently a measure of the external work
done by the xmit mass of gas in expanding against the pressiu*e
p, while it is being heated through a range of 1 degree.
Assiune that we have a given mass of gas m, whose volume
is V at 0°C., confined in a cylinder by a piston of area A, imder a
pressure p, per unit area. If the piston be perfectly free to move,
and heat be applied bringing about an elevation of temperature
T, the pressure being maintained constant during the process,
then the volume will be increased by an amount vaT. The dis-
tance through which the piston moves during the expansion is
vox/ A ; and the external work done, which is numerically equal
to the product of force and displacement, is
W^pA-j-^pvan (16)
The heat consumed in doing the external work, expressed in
mechanical imits, is
H=Jm{Cj,-C,)'z; , . (16)
FUNDAMENTAL EQUATIONS OF GASES 49
where J is the mechanical equivalent of heat. But, imder the
assumed conditions, H and W are nimierically equal; hence, from
equations (15) and (16), we have
Jfn{Cp — C,)t = pv(xt ;
from which
Equation (17) * enables us to compute the mechanical equiv-
alent of heat from the known constants of a gas. The constants
of dry air are as follows: Cp = 0.2376, C,= 0.1690, a =0.003665,
and the mass of 1 c.c. of air at 0°C., under a pressure of 1.01325
XlO^' dynes per sq.cm., is 0.001293 grams*. Substituting these
values in equation (17), and assuming 1 c.c. for the initial voliune,
we find
. 1.0132 X10^X0.003665 . mos^iA? i •
■^= 0.001293(0.2375-0.1690) ^^-^^^^^^ ^'^ ^' ^"^ "^^™'
The value of J thus obtained differs by a small percentage from
that obtained by the direct conversion of mechanical work into
heat. We have here then a complete verification of the numer-
ical relation between heat and work. That is, in the one case,
mechanical work is directly converted into energy, in the form
of heat, and the numerical ratio of the two quantities involved is
determined. In the other case, heat is converted into work, and
the numerical relation between the two quantities is again
determined; and the difference, between the two values so deter-
mined, is well within the limits of observational error.
* It was by this method that J. R. Mayer first computed the mechanical
equivalent of heat. Various writers have attempted to take some of the
credit from Mayer, by asserting that it was not then known that no energy
is required for the simple expansion of a gas. But Gay-Lussac had per-
formed this experiment, and anyone reading Mayer's original papers will
see that he was aware of this and interpreted the experiment properly.
50 HEAT
48. Adiabatic Equation. If a gas is compressed, work is being
done on it and heat is necessarily developed; and since the pres-
sure of a gas, other things being equal, rises with the temperature,
it follows that, unless the heat developed by the compression is
abstracted from the gas as rapidly as it is developed, the pressure
must rise more rapidly, with respect to the amoimt of compres-
sion, than it would during isothermal compression. There are
certain processes where compression and dilatation take place
■so rapidly that there are practically no exchanges of heat between
the various parts of the system. Changes, during which no heat
enters or escapes, are called adiabatic changes.
A good example of adiabatic changes are the compressions
and rarefactions which take place in a medium when a soimd
wave exists in it; the time of compression, or rarefaction, being so
small that practically no heat is transferred from particle to
particle.
In general, if we are dealing with a unit mass of a perfect gas,
we may write
dQ^CdT+pdv] (18)
where all quantities, of coiu*se, are expressed in the same units.
dQ, expressed in mechanical units, is the quantity of heat absorbed
by the gas, or else abstracted from it, C^dT is the quantity of heat
involved in bringing about the change of temperature dr, and
pdv represents work done either by the gas or on the gas.
As a matter of illustration, assume work is being done on the
gas in a manner such that its temperature rises and that there is
also heat given to the surroundings. If we consider work done
by the gas and heat absorbed by the gas positive, then, in equation
(18), if applied to this case, both dQ and pdv become negative;
on the other hand, Cj,dT remains positive.
If, now, during the change in volume, no heat enters or escapes,
the process will be adiabatic; and equation (18) becomes
C,dr+pdt;=0 (19)
FUNDAMENTAL EQUATIONS OF GASES 61
In equation (19), if dv is positive, dT must be negative; i.e., if
external work is done by the gas it is done at the expense of the
intrinsic energy of the gas, and the temperature must fall. Like-
wise, if work is done on the gas, since according to our assumption
no heat escapes, its temperature must rise. To solve equation
(19) we will substitute for dT its value, as found from equation
(9), Art 43; i.e., by differentiating
pv=RT,
we find
dT=^P^^ (20)
Now, R can be expressed in terms of the two thermal capacities.
To find this relation, assume that we are dealing with a imit mass
of the gas and allow it to expand under a constant pressure p,
while its temperature is increased by unity. If vi is the initial
and V2 the final volume, the external work done is
p(«2— t;i) = Cp— Cf.
It being understood that all quantities are measured in mechan-
ical units. But, from equation (9), it follows directly, that
p{v2-vi)=R(T2-Ti);
and, since by the conditions we have T2'-Ti equal to unity, it
follows that
Substituting this value of i2 in equation (20), and the value of
dT so obtained in equation (19), we find
CPp±^+pdv=^0;
from which
t) V+f -O- m
62 HEAT
But, as previoiisly explained, CplCt is practically a constant.
Representing this constant by n and substituting in equation
(21), we have
dv .dp ^
n — h— =0;
from which, by integration,
logv'^+logp^fci;
where ki is the constant of integration. Or, expressed in another
form,
logp»"=fci;
from which
pv^'^k; (22)
where X; is a constant depending upon the units chosen. Equation
(22) gives the relation of pressure and volume for a gas, during
adiabatic changes, and is known as the adiabatic equalion.
General Equations of Gases
49. The change of heat involved when a gas suffers a change
is a fimction of the temperature, pressure, and volume; i.e.,
Q=f{T,p,v).
But, since any two of these quantities may vary independently
of the third, we may write
Q=/(r,p), Q=f{T,v\ and Q^rip.v).
By partial differentiation of these functions, we obtain
«-{ii).^''+(f) *■ • •. • <^)
«-(w)5+©,*- • • • • <«'
and
'^^=(if).'^+(S),*'-- • • • • (26)
FUNDAMENTAL EQUATIONS OF GASES 53
If it be assumed that we are dealing with a iinit mass of gas^
then in equation (23), we have
m
since in this case, (^) is the thermal capacity per unit mass
at constant pressure. In a like manner,
(
IV/T
is also a imit thermal capacity, and is the ratio of change in heat
to change in pressure at constant temperature. Substituting,
in equation (23), we have
dQ=Cj,dT+mdp (26)
In equation (24)
(g)dr=WT;
since (■^) is the thermal capacity per imit mass of the gas at
constant volimie. In a like manner.
(
IV/T
is a unit thermal capacity, and is the ratio of change in heat to
change in volume at constant temperature; hence, by substi-
tuting in equation (24), we find
dQ^C,dT+ldv (27)
Again, in equation (25) the two quantities, viz, (t" ) > ^^^ ratio
of change in heat to change in pressure at constant volume, and
( ^) , the ratio of change in heat to change in volume at constant
64 HEAT
pressure, are both unit thermal capacities. If we represent the
former by j, and the latter by o, then equation (25) becomes
dQ=-jdp+odv (28)
Since all of these equations must be true for the particular
case when the left-hand members are equal, the right-hand mem-
bers of equations (26) and (27) may be equated, and we have
Cj,dT+mdp = CpdT+ldv (29)
From the fundamental statement
v^fiPy r),
we have
*-(|)*+(r.).^^-
Substituting this value of dv, in equation (29), we have
and
C,dT+mdp=lc,+l(^^\dT+l(^^Jlp. . . (30)
Since this equation is true when the corresponding changes
in the two members are equal, it follows that
and
^°it\ (32)
Again
from which
P=/(f, T);
*-© *+(ll)5- (»>
FUNDAMENTAL EQUATIONS OF GASES 55
Substituting the value of dp as given in equation (33), in equation
(29), we have
from which
Cp+m
(i).}'^^+'"(S)r*'=^'^^+^'
hence, by equating like coefficients, we find
C,-a=-m(||)^ (34)
By equating the right-hand members of equations (26) and (28),
we find
CpdT+mdp=jdp+odv (35)
From the fundamental statement
T=f(v, V),
we have
'''-(l7)*+(lf)*^
and substituting this value of dT, in equation (35), we obtain
hence, by equating like coefficients, we find
From equations (27) and (28), we have
CvdT+ldv =jdp + odv ;
56 HEAT
and, by substituting for dT its value, we find
from which
We then have the following values:
and
•'(f).--'
From the characteristic equation .
■pv=RT,
we find
hence
(lv\ ^R
i=|(C,-C.) (38)
Again
hence
m=~(C,-C,). " (39)
FUNDAMENTAL EQUATIONS OF GASES 57
Also
.J
hence
R
Finally
i=^C^ (40)
(ir\ =e.
,Bo/, R'
and
o=|Cp (41)
Substituting in equations (26), (27), and (28) the values of I, m,
j, and as just determined, we find
dQ = CpdT--iC„-C,)dp, (42)
dQ = C,dT+^{Cp-C.)dv, (43)
and
dQ=^C.dp+|c,(fo. (44)
These equations, viz, (42), (43), and (44), may be put into
different forms; since, from the characteristic equation
pv=-RT,
we have
T/p=v/R, and p/R=-T/v.
From the assumption, then, that in the fimdamental statement
(3=/(r, p, t;),
58 HEAT
any two of the variables may vary independently, while the third
is maintained constant, we have obtained three distinct equa-
tions, viz, equations (42), (43), and (44). It is of further interest
to note that if in any one of these three equations the right-hand
member is equated to zero, the adiabatic equation is obtained.
Assuming that the change in equation (44) is adiabatic, then
dQ = 0, and we have
Ctvdp^—Cppdv;
from which
and
Cpdv dp^
»/*= -If'
from which
n log v= —log p+ki.
Or, expressed in another form
and, finally
pV'^k (46)
This may also be obtained from the other equations, as may be
readily shown. From
we have
^m_ pdv+vdp
Substituting this value of dT in equation (43) and equating to
zero, we find
Ct{pdv+vdp) = '-p(Cp—Cv)dv;
from which
Cvvdp^—Cppdv;
FUNDAMENTAL EQUATIONS OF GASES 59
which is identical with the result obtained from equation (44)
under the same assumption. Again, substituting in equation
(42), for iT its value and equating to zero, we find
from which
which is again identical with that previously obtained imder the
same assimiption.
If it be desired to find the temperature of a gas, corresponding
to a given pressure and volimie, during an adiabatic change, in
terms of the initial temperature and pressure and the given pres-
sure, or in terms of the initial temperature and volume and the
given volume, we proceed as follows: Let Ti, pi, and v\ be, re-
spectively, the initial temperature, pressure, and volume, p and
r, respectively^ the pressure and volume, for which the corre-
sponding temperature, T, is sought. Then, since the two points
are on the same adiabatic, we have
Pivi''=7w'*; (46)
and from the characteristic equation, we have
pit;i=ii!ri, (47)
and
3W=i2r (48)
Substituting in equation (46), the values of v\ and v, as found from
equations (47) and (48), we find
n-l
2'=ri(^) " (49)
60 HEAT
In a similar manner, by substituting in equation (46), the values
of pi and p, as found from equations (47) and (48), we obtain
t-tM:-!
(50)
Vapors
50. Vaporization. The gaseous states of bodies, which under
ordinary conditions of temperature and pressure are either liquids
or solids, are called vapors; and the process by which the vapor
is formed is called vaporization.
In general, vaporization takes place in two distinct ways. In
the one process, called evaporation, vapor is continually formed
at the exposed surfaces of liquids; and in the other process, called
ebvUition, bubbles of vapor are formed in the body of the liquid
or at the heated surfaces.
61. Evaporation. If a liquid be enclosed in a space, only
part of which is occupied by the liquid, then vapor immediately
forms and occupies the space above the liquid. This continues
imtil the vapor has reached a certain density which depends
upon the temperature, and is greater as the temperature is higher,
but is always the same for the same temperature. In other
words, for any given temperature there is a maximum density
and hence, a maximum pressure, which the vapor is capable of
exerting. When this state is reached the vapor is said to be
saturated. That is, for the given temperature the space contains
the maximiun possible amount of vapor. If, after this state has
been reached, the temperature be maintained constant and an
attempt be made to increase the pressure by the application of
an external force the result will be, not an increment in pressure,
hut a diminvJtion in vdlume, at constant pressure, and a corre-
sponding amount of condensation. In other words, the pressure
for a saiuraied vapor ai constant temperature is constant. Or,
FUNDAMENTAL EQUATIONS OF GASES 61
to put it in still another way, the temperature of a saturated
vapor is uniquely defined by its pressure.
62. Addition of Vapor Pressures. The rate of evaporation
depends, of course, upon the rate at which heat is being supplied;
but, as has just been stated, the final pressiu*e reached depends
merely upon the temperature. Furthermore, evaporation takes
place more rapidly in a vacuum than in a space occupied by the
vapor of some other substance; however, the final pressure reached
by the vapor will be almost, though not quite, as high, when the
space is partially occupied by some other gas or vapor, than it
would be were the space originally a vacuum, provided always,
that the temperature be the same and that there be no chemical
action between the vapors. This statement was first made by
Dalton, viz, when evaporation takes place in a space filled by
another gas, which has no action on the vapor, the final pressure
reached by the mixture is equal to the sum of the pressures of its
, constituents. Careful experiment shows Dalton's statement to
be approximately, though not rigidly true.
63. EbuUition. As has been previously stated, when heat is
applied to a liquid, the temperature rises progressively with
continued application of heat until a certain point, which depends
upon the pressure, is reached, when the temperature remains
constant. This is the boiling-point for the given pressure; and
is that temperature for which the pressure of the vapor is equal
to the superimposed pressure. Since the pressure at any point
in the liquid, is equal to the pressure at the surface plus the pres-
sure due to the liquid, from the surface to the point under con-
sideration, it follows that, the temperature varies for different
depths below the surface of the liquid. Hence, the temperature
of the boiling liquid is not a constant throughout; but increases
slightly with the depth.
When equilibrium has been attained, i.e., the temperature
becomes constant, then all the energy that is supplied, in the form
of heat, is consimied in converting the liquid into a vapor. This
62 HEAT
energy oonsists of two parts, viz, one part being that energy
which is required to overcome the inherent forces, that is, to
separate the particles so as to form vapor, and the other part
to overcome the external pressure during the augmentation of
volume. The former is called the heat of disgregation and the
latter the heat of expansion.
I Pressure, however, is not the only factor that fixes the boiling-
point of a liquid. As examples, the following may be cited:
The nature of the material of the containing vessel has some influ-
ence. If the liquid be first carefully freed from the imprisoned
air, the temperature may be raised considerably above the tem-
perature at which ebullition ordinarily takes place. Impurities
in the liquid influence the boiling-point. And finally, salts dis-
solved in ajiquid always raise the boiling-point. As an example,
the boiling-point of a saturated solution of water with conmion
salt is about 109°C. But the temperature of the saturated vapor
of a liquid is always the same for the same pressure, no matter
what the temperature of the liquid. It is for this reason tha.t the
temperature of steam, rather than that of water, imder a pressure
of one standard atmosphere haa been chosen as the boiling-point.
64. Critical Temperature. When a liquid is heated in a closed
vessel the vapor accumulates above the liquid and augments
the pressure. Up to a certain point, differing for different liquids,
there is a sharp definition between the liquid and vapor; but, for
every liquid there is reached, finally, a temperature when this
definition ceases, and the liquid disappears and is completely
converted into vapor, even though the volume occupied by the
vapor is but little greater than that occupied by the liquid. The
temperature at which this takes place is called the critical tem-
perature for the substance. And, it appears that, for temper aiures,
higher than this, no meatier what the applied pressure, the substance
can exist only in the gaseous state. The following table gives a few
substances together with their approximate critical temperatures,
and the corresponding pressures:
FUNDAMENTAL ^EQUATIONS OP GASES
63
Subfltanoe.
Temperature in
Degrees, C.
Pressure in
Atmospheres.
Carbon-dioxide
31
156
194
365
-118
-146
-234
77
79
36
195
50
33
20
Sulphur-dioxide
Ether
Water
OxvKen
Nitrofcen
HvdroKen
**rf *** v©**** •••
66. It is interesting to note that there apparently is a relation
between heat of disgregation of a substance and its critical tem-
perature. Let ff be the specific volume of the liquid; i.e., the
volume occupied by unit mass of the liquid, and 8 the specific
volimie of the dry saturated vapor, then the increment in volume,
when a unit mass of a liquid is converted into vapor, is
/i=«-ff;
(51)
where fi is the increment in volume. The external work, or the
heat of expafision, is
Tr=p/t=p(s-<j); (62)
where p is the pressure during the evaporation. To express the
heat of expansion, during evaporation, in thermal units, we must
divide by J, and equation (52) becomes
Tr=ilp(s— a);
(53)
where A = l/J.
If we represent by r the heat of vaporization, and by p the
heat of disgregation, equation (53) becomes
from which
r-p=ilp(«-<j);
p=r— ilp(«— <j).
(54)
(55)
64 HEAT
To illustrate, we will now compute the heat of expansion for
water when it is converted into dry steam at lOO^C.
Assume that we are dealing with 1 gram of water at 100°
C. Its volume will be approximately 1 c.c; and the volume of
its saturated vapor imder atmospheric pressure will be about
1670 c.c. Substituting, in equation (54), for «— a, A, and p the
numerical values, we find
1.013 X10«X 1670* .^^ , .
r— p= A lQg>yi0 7 =40.3 gram calories per gram.
This gives for the heat of disgregation, which is the difference
between the heat of vaporization and the heat of expansion,
536.5-40.3=496.2.
Zeimer gives an empirical equation, for the heat of disgregation
for water, which gives results very close to those obtained by
equation (55) ; this equation is
p = 575.4 -0.791t; (56)
where p is in gram calories per gram and t in degrees centigrade.
In general, the heat of disgregation becomes less as the tem-
perature becomes higher; and the critical temperature appears
to be that for which the heat of disgregation becomes zero. In
the case of some liquids, very close agreement is found between the
values for the critical temperatures, as found by direct experiment,
and those values calculated from the empirical equations for the
heat of disgregation. In other cases, again, there are discrepancies
of considerable magnitude. A notable case is that of water;
but, it must be remembered that the critical temperature of water
is very high, and therefore, its determination is attended by dif-
*8 not being known definitely to the fourth significant figure, it is
immaterial whether we use 1670 or 1669; or, in other words, the volume
of the liquid, in this case, is negligibly small in comparison with that of its
vapor.
FUNDAMENTAL EQUATIONS OF GASES 65
ficulties. Furthermore, the heat of vaporization for water has
not been determined for such high temperatures; hence, the empir-
ical equation for such ranges is doubtful.
56. Total Heat of Steam. Before leaving the subject of satu-
rated vapors, we will give, on account of the importance in steam
calculations, the empirical equations for the total heat of satu-
rated steam, and the heat of vaporization. By total heat of steam
is meant the quantity of heat required to raise the temperature
of unit mass of water, from the melting-point of ice, to the tem-
perature imder consideration and convert it into saturated steam
at that temperature. In the French system the total heat is
given by the equation
fl = 605+0.305 T gram calories per gram. . . . (57)
In the English system the total heat is given by the equation
ff = 1082 +0.305 T B.T.U. per pound (58)
In equation (57), H equals the quantity of heat, in gram calories,
required to raise 1 gram of water from 0°C., to the temperature
T°C., and convert it into a saturated vapor at that temperature.
In equation (58), -ff equals the quantity of heat, in British thermal
units, required to raise 1 pound of water from 32°F. to t^F.,
and convert it into a saturated vapor at that temperature.
67. Heat of Vaporization for Water. The empirical formula,
which gives the heat of vaporization of water, for the various
temperatures, is
r=1114-0.7TB.T.U. per pound. . . . (59)
Equation (59) is not quite as accurate as equation (58) ; but
for most steam calculations it is sufficiently precise; since, by
means of it the heat of vaporization is foimd, with an error of less
than 1 per cent, between lOO'^F. and 400**F.
66 HEAT
68. Superheated Vapors. Any vapor which, for a given
pressure, is at a temperature higher than that corresponding to
saturation, for the given pressure, is said to be superheated. This
is only possible when the vapor is not in contact with its own
liquid. Vapors which have been superheated obey Boyle's law
approximately; and furthermore, for adiabatic changes, the
equation
holds; n having different values, depending upon the vapors
with which we are dealing.
69. Hygrometry. Hygrometry has for its object the deter-
mination of the state of the atmosphere with respect to the aqueous
vapor present. The amount of aqueous vapor present in the
atmosphere is a very variable quantity. The effect of the vapor,
however, depends not only upon the quantity present, but also
upon the temperature. These two facts are included in the single
statement, that the effect of the vapor depends upon the relative
humidity.
By the expression relative humidity, is meant the ratio of the
actual density of the vapor, contained by the air, to the density
which it would have if there were saturation for the given tem-
perature. Or, if expressed as a percentage, the relative humidity
is the percentage of saturation for the given temperature.
60. Dew Point. The dew point is that temperature at which
the vapor present in the atmosphere, begins to condense; i.e.,
the point of saturation. Assume that a certain portion of the
atmosphere is cooled until the vapor present begins to condense.
The temperature at which this takes place is readily found by
experiment. By referring to a curve giving the relation of
temperature and pressure of saturated steam, we can readily
find the pressure corresponding to the dew point. Designating
this pressure by pi, and by p2 the pressure of saturated vapor
corresponding to the temperature of the atmosphere, then since
FUNDAMENTAL EQUATIONS OP GASES 67
non-saturated vapors approximately obey Boyle's law, the density
of the vapor present in the atmosphere is to the density the vapor
would have were there saturation, very nearly, as pi is to p2.
Hence, we have approximately, for the relative humidity
A = 100^ per cent.
To illustrate further, we will consider a concrete case. Assimie
the temperature of the atmosphere to be 25°C., and that of the
dew point 15°C. From the steam curve we find pi = 1.278 cm.
of mercury, and p2= 2.369 cm. of mercury. From this, the rela-
tive hmnidity, expressed as a percentage, is foimd to be
1 278
A = 100X^|g=53.9 per cent.
61. Absolute Humidity. The amoimt of moisture, expressed
in grams, contained by a cubic meter of air is called the absolute
humidity. This is found in a very simple manner. The dew
point is determined, givmg the temperature of the saturated
vapor, and from this, by referring to steam tables, the mass per
unit volume is foimd.
CHAPTER VI
ELASTICITIES AND THERMAL CAPACITIES OF GASES
62. The adiabatic equation, for gases and vapors, being so
frequently employed in the discussion of the theory of heat
motors, it is important to say a few words about the determination
of the ratio Cp/Cv=^n. The determination of thermal capacities
of gases is attended by far greater difficulties, than those found
in the determination of thermal capacities of liquids and solids.
This is due to the fact that the thermal capacity of a gas is always
very small in comparison with that of the containing vessel.
63. Thermal Capacity at Constant Pressure. Regnault was
the first to determine accurately the thermal capacities of gases
under constant pressure.* The method is essentially as follows:
The gas, whose thermal capacity is sought, is contained in a large
reservoir, imder a high pressure, from which it is passed through
a spiral tube immersed in a bath, the temperature of which is
maintained constant. The spiral tube is of sufficient length to
insure the gas leaving it to be at the same temperature as the
bath. In the tube, connecting the reservoir with the spiral tube,
is a valve, by means of which the pressure of the gas is maintained
constant. A second spiral tube, through which the gas must
pass, is immersed in a calorimeter filled with water; the thermal
capacity of the calorimeter and contents being known. The spiral
tube, immersed in the calorimeter, is of such length that the tem-
perature of the gas, throughout the progress of the experiment,
is reduced to that of the calorimeter before being discharged into
* For a full description, see Preston's "The Theory of Heat," Chapter IV,
Section VI.
68
ELASTICITIES AND THERMAL CAPACITIES OF GASES 69
the atmosphere. From the initial and final pressure of the gas
in the reservoir, together with its temperature, which is main-
tained constant, by means of a suitable bath, the mass of gas pass-
ing through the calorimeter, during the progress of the experiment,
is readily found. Furthermore, from the thermal capacity of
the calorimeter and contents, together with the initial and final
temperatures, the quantity of heat absorbed by the calorimeter
during the progress of the experiment, proper corrections being
made for losses, is determined. And, since the temperature of
the gas before entering the calorimeter, as well as the average
final temperature, is known, and also the mass of gas which has
passed through the calorimeter, the thermal capacity per imit
mass is determinate.
64. Thennal Capacity at Constant Volume. In the experi-
ment just described the quantity of gas employed is not limited
by any containing vessel; for, the reservoir in which the gas is
contained may be of any size whatsoever, without having any
influence on the result. Therefore, a large quantity of gas may
be used, and consequently, a considerable range of temperature
may be obtained in the calorimeter. However, when it is desired
to determine the thermal capacity of a gas at constant volume,
the quantity of gas upon which we are experimenting, is limited
by the containing vessel; and the thermal capacity of the con-
taining vessel is always large in comparison with the thermal
capacity of the enclosed gas.
66. Joly's Steam Calorimeter.* In the most primitive form,
the steam calorimeter consists of the pan of one side of a beam
balance placed in an enclosure with the specimen, whose thermal
capacity is sought, supported by the pan. When steam is admit-
ted into the enclosure, condensation takes place until the temper-
ature of the test specimen is equal to that of the steam. The
steam then passes through the enclosure without further conden-
For a complete description of this apparatiis, see Preston's "The Theory
of Heat," Chapter IV, Section V.
70 HEAT
sation. When this condition has been reached, the balance is
counterpoised and the mass of condensed steam, which has been
collected by the pan, is noted. From the initial and final temper-
ature of the test specimen, together with the quantity of water
collected by the pan, and the heat of vaporization for this par-
ticular temperature, the thermal capacity of the test specimen is
readily found.
66. Differential Steam Calorimeter. In this form, both pans
of the balance, which are made so that they have equal thermal
capacities, are suspended in the enclosure. On the one pan is
placed a spherical vessel, which has been exhausted, and on the
other pan a spherical vessel of like dimensions and equal thermal
capacity, filled with the gas whose thermal capacity is sought.
When steam is now admitted into the encloito'e, the quantity of
water which collects jn the pan, supporting the vessel containing
the gas, is greater than that which is collected in the pan supporting
the exhausted vessel. This is necessarily so; since the vessel,
together with the contained gas, has a thermal capacity greater
than the exhausted vessel. From the excess of condensation in
the'one pan over that in the other, which is obtained directly
by weighing, together with the initial and final temperatures
of the enclosure and the mass of gas contained by the one vessel,
the thermal capacity per unit mass of the gas at constant volume
is readily foimd. Correction, of course, being made for change
in volume of the containing vessel for change in temperature.*
It must, of course, always be remembered that it is impossible
to obtain absolutely accurate results by this method; since the
thermal capacity of the gas is always small in comparison with
that of the containing vessel. Still, Dr. Joly, who is the inventor
of this method, has obtained fairly good results.
67. Method of Clement and Desoimes. In this method the
gas, whose thermal capacity is sought, is contained in a large vessel
For a complete description, see "The Theory of Heat," by Preston,
Chapter IV, Section V.
ELASTICITIES AND THERMAL CAPACITIES OF GASES 71
provided with a delicate manometer. When the contained gas
has assumed the temperature of the surroundings, its pressure,
which must diiBfer from the atmospheric pressm'e, is carefully
ascertained. When this has been done, a stop cock, having a
large orifice, is opened and then closed after a very short interval
of time. The time which elapses between the opening and closing
of the stop cock must be so small that the change in the gas may
be assumed adiabatic. During this change the temperature
changes; i.e., there will be, either an elevation of temperature,
if the pressure in the flask was initially less than the atmospheric
pressure, or else, a diminution of temperature, if the pressiu*e in
the flask was initially greater than the atmospheric pressure.
After the vessel and contents have again assumed the initial
temperature, viz, the temperature of the surroimdings, the pres-
sure is again carefully noted.
If, now, we denote by pi the initial pressure of the gas, by vi
the corresponding volume per unit mass, by p the atmospheric
pressure, which is also the pressure of the gas when the stop cock
is open, and by V2 the volume per unit w>ass after the stop cock is
closed, then since the change is assumed adiabatic, we may write
PlVi'*=pV2'' (1)
Also, since the initial and final temperatures are the same, we have
PlVl=P2V2; (2)
where p2 is the pressure in the vessel after the temperature of the
surroundings has again been assumed.
From equation (1), we find
\W Pi'
from which
^^log (p/pi) .
log (t;i/t;2)
72 HEAT
From equation (2) we find
from which
log^=Iog2?.
V2 Pl
Substituting in equation (3) for log (viM), its value, log (P2/P1),
we obtain
^^log( p/pi)
log (P2/P1)
Or, expressing this in another form, we have
^^ logp-logpi .g.
logp2-logpi
Since p, pi, and p2 are known, n is determinate. In this
manner. Roentgen found for dry air the value n = 1.405.
This method is open to criticism, in so far that when the stop
cock is opened, oscillations occur; and it does not necessarily
follow that, at the instant of closing, the pressure in the vessel is
equal to that obtaining outside.
68. Isothermal and Adiabatic Elasticities. The ratio of the
two thermal capacities of a gas is most accurately found by
determining the speed of propagation of a disturbance through
the gas. We will first show that the ratio of the two thermal
capacities is numerically equal to the ratio of the two elasticities.
From the statement of Boyle, we have, the temperature being
maintained constant,
pv=fci; (6)
where fci is a constant, depending upon the units chosen. By
differentiation, we find immediately
pdv+vdp=0;
ELASTICITIES AND THERMAL CAPACITIES OF GASES 73
from which
-^-p; (7)
-(t)
c
the minus sign denoting merely that the volume decreases as the
pressure increases. Now, the left-hand member of equation ^
(7), is numerically equal to the ratio of change in imit stress to
the corresponding change per unit volume; and is, therefore,
by definition, the expression for the modulus of elasticity. Hence,/
for a gas obeying Boyl^a law, the elasticity is numerically equal
to the pressure.
If now, we take the adiabatic equation, viz,
pt;**=fc2, (8)
where k2 is again a constant depending upon the units chosen,
and differentiate, we find
»-i>^w7..^n.
from which
and
v^dp+nv^ " pdv = 0;
vdp+npdv=0;
^ np (9)
-(?)
The left-hand member of equation (9) again expresses, accord-
ing to definition, the modulus of elasticity. Hence, the modulus
of elasticity when no heat is allowed to enter or escape, i.e., for
adiabatic changes, is numerically equal to the product of the ratio
of the two thermal capacities and the pressure.
Denoting the isothermal elasticity by Et, and the adiabatic
elasticity by Eh, we have
Eh_np_ . .
Erj"" ^^°^
74 HEAT
From equation (10) we see that the ratio of the two principal
elasticities is the same as the ratio of the two principal thermal
capacities.
69. Propagation of Wave Motion in an Elastic Medium. To
properly appreciate how the ratio of the two elasticities, and hence
the ratio of the two thermal capacities, of a gas is fomid from the
speed of propagation of sound in the gas, it is essential to study
the character of the motion by means of which sound is propagated
in an elastic medium.
Let ABf of Fig. 5, be a prism, of indefinite length and constant
cross-sectional area, filled with a homogeneous elastic medium;
and let the piston P have impressed upon it a constant acceler-
ation toward the right. If the medium had absolutely no inertia,
or were perfectly rigid, then the whole substance, between A and
— IT
1
i|
1
1
1
^1
1
1
1
^1
1
1
1
i|
1
1
1
— MIL
Fig. 5.
Bj would suffer precisely the same displacement in a given interval
of time. Due, however, to the inertia, the layer next to the piston
will be compressed; the pressure of this layer now being greater
than that of the medium in the imdisturbed condition, it will
react upon the second layer and compress this, which again in
turn compresses the third layer, etc. Finally, when every layer
throughout the prism, has been compressed by an amount such
that its internal pressure is precisely equal to the applied pressure,
then the acceleration of each layer will be the same and equal
to that of the piston. .
Assume now, that the piston P is caused to vibrate periodically,
with a small amplitude s. In tracing out a vibration we will
begin by assuming the piston in the neutral position, moving
toward the right, and the pressure of the medium, throughout,
ELASTICITIES AND THERMAL CAPACITIES OF GASES 75
the same as that m the undisturbed condition. As the piston
moves toward the right, condensation takes place in the medium;
the condensation being greatest for the layer in contact with the
piston and becoming less as the distance from the piston increases.
Suppose now, that when the piston has reached its maximum difh
placement 8 toward the right, the wave of condensation has reached
the section represented by a; i.e., the pressure of the medimn
at the section a is the same as that in the undisturbed condition,
and greater for all portions to the left of a. As the piston now
begins to move toward the left, the wave of condensation continues
moving toward the right; but, the pressure behind the piston
begins to decrease, and by the time the piston has again reached
its neutral position, the pressure of the medium directly in contact
with the piston is the same as that in the undisturbed condition.
The wave of condensation will, in the meantime, have traveled
to the section h ; the distance ab being equal to Aa. The maximum
condensation is now at a, and tapers off to zero from a to & and
from a to A. As the piston now continues moving toward the
left, the medimn behind it becomes rarefied; and a waoe ofrar^
faction travels toward the right. By the time the piston has
reached its extreme left-hand position, the wave of rarefaction
will have reached the section a, and the wave of condensation the
section c; where the distances he and ab are equal. The maximum
condensation now exists at b, and the maximum rarefaction at the
piston. As the piston now begins its journey toward the right,
the pressure behind it begins to rise, imtil it reaches the neutral
position, when the pressure at the piston is equal to that of the
medium in the undisturbed condition. In the meantime, the wave
of condensation has traveled to the section d, where the distances
cd and he are equal. Hence, during the time required by the piston
to complete a period, the disturbance has traveled from A to d;
and the conditions now existing are: Maximum condensation
at c, maximum rarefaction at a, and at A, b, and d, the pressure
is equal to that of the medium in the imdisturbed condition.
76 HEAT
As the piston now continues moving toward the right, a wave
of condensation moves toward the right from Aj and also from
d; and by the time the pistoB has completed its second cycle,
the disturbance will have traveled to h; where the distances
dh and Ad are equal. The condition of the medium between A
and d is now at every section precisely the same as it was at the
end of the first cycle. Also, the condition of the medium between
d and h is precisely the same as it is between A and d; i.e., the
condition at e is the same as at a, at / the same as at b, at g the
same as at c, etc. At the end of the third cycle the disturbance
will have traveled to the right of A, a distance equal to Ad=dh;
and the condition of this portion will also be the same as the con-
dition of the portions from A to d and from d to h. At the end
of N cycles, the disturbance will have traveled through a distance
equal to the product of N and Ad.
The distance through which the disturbance travels while
the piston goes through one cycle is called a wave length, and repre-
sented by the letter X. Or, in other words, this is the distance a
disturbance travels before conditions are beginning to be exactly
reproduced.
From the previous discussion, it is obvious that if there be
performed N vibrations per imit time, and X is the wave length,
then the speed of propagation is given by
S^N-k (11)
It is important to note that the distance through which the
disturbance travels during a cycle depends upon the time consimied
in performing that cycle; i.e., if the frequency — the number of
vibrations per unit of time — ^be increased, then according to the
discussion, the wave length will be proportionately less, such that
the product of wave length and frequency is constant. This is
fully verified by experiment for the speed of propagation of soimd
in gases.
ELASTICITIES AND THERMAL CAPACITIES OF GASES 77
70. Speed of Propagation in Terms of Elasticity and Density.
Assume, as in Fig. 6, a cylinder of indefinite length and constant
cross-sectional area A, filled with a homogeneous elastic medium
whose density is p, and pressure per unit area in the undisturbed
condition p. Assume further the frictionless piston P, having
applied per unit area a pressure p+Jp; where Jp is a small frac-
tional part of p.
Now, as a matter of convenience, assume that the prism, repre-
sented in Fig. 6, is divided into unit lengths, 1, 2, 3, etc., up to N;
where N represents the distance the disturbance travels in a
time t. The effect of the application of a pressure to the piston,
in excess of the pressure of the medium, will be twofold; i.e., the
medium will be compressed and also set in motion. It is evident
that when any element of the medium in the prism has reached
p
I
P+AP
2
8
N
As
Fig. 6.
a pressure per unit area equal to p+Jp, it cannot be further com-
pressed, but will merely serve to transmit the applied stress to
the next element.
Let Ja be the amount of shortening a unit length undergoes
while its pressure rises from p to p+Jp. The total shortening
then, that the prism of length N undergoes in being compressed
from the pressure p to a pressure p+Jp, and consequently the
distance through which the piston moves during the time this
change takes place, is
and, since the time consumed to bring about this change is t, the
speed with which the piston has been moving is
d NJs
t t
(12)
78 HEAT
But, at the instant that the pulse has passed through the distance
iV^all that portion of matter included in the length iV,of the prism,
is moving with a speed the same as that of the piston as given by
equation (12); hence, its kinetic energy is
m' »'>
Since the total change in volume is AN As, and the average
resisting pressmre per unit area is p+ Jp/2, the work, due to com-
pression, is
W2^{ANA8){v+^^ (14)
But the total work done on the system must be equal to w\-\'W2\
hence;
from which
m-y
Ap
(15)
Now, N/t^Sf the speed of propagation; and Jp/Js, in the
limit, represents the ratio of unit stress to unit strain, and hence,
is equal to (x, the modulus of elasticity. Substituting, in equation
(15), we have finally
s=^/^; (16)
i.e., the speed of propagation of a disturbance through an elastic
medium is numerically equal to the square root of the ratio of
elasticity to density.
The speed of propagation of soimd in air is very readily deter-
mined by experiment; and is found to be, at 0°C., very nearly
332 meters per sec. The vibrations in a sound wave take place
ELASTICITIES AND THERMAL CAPACITIES OF GASES 79
so rapidly that the changes are sensibly adiabatic; hence (i, in
equation (16), will be replaced by the adiabatic elasticity, and we
have
from which
.-fS (17)
Substituting numerical values, in equation (17), we find
_ (33,200 )2 X0.001293 _
^" 1.0132X10« -A.4U0i-.
The value 1.405 is generally used for dry air; but, for most
practical purposes, 1.4 is sufficiently close.
CHAPTER VII
PROPAGATION OF HEAT
71. Heat is transferred from one place to another in three
distinct ways, viz, by radiation^ by corwection, and by conduction.
72. Radiation. In Chapter II, we dealt with Newton's law
of cooling, without considering in what manner the cooling takes
place. As a matter of fact, in the cases considered, the pooling
was due to two distinct phenomena. To illustrate this, we will
consider a concrete case, viz, an incandescent lamp, which con-
sists of a filament inside of a glass bulb; the bulb having been
exhausted, so that the filament is practically in a vacuum. The
propagation of heat from the filament to the glass bulb, that
is, through a vacumn, is called radiation; or, in other words,
radiation is the propagation of heat through space without the aid
of any maierial substance. On the other hand, the dissipation of
heat from the surface of the bulb is due, not only to radiation,
but also convection; and the propagation of heat from the inner
siuf ace of the bulb to the outer surface is due to conduction. The
propagation by convection and conduction will be considered
later.
73. Theory of Ezchanges. Provost, in 1792, promulgated
the theory that there is a continual exchange of heat between
bodies, even when they are at the same temperature. Provost's
theory may, perhaps, be best explained by means of the following
illustration: Suppose a body suspended in a vessel, which has been
completely exhausted. Assume further, that the walls of the enclo-
sure are maintained at a constant temperature, and that the body,
when first placed in the enclosure, has a temperature higher than
80
PROPAGATION OF HEAT 81
this. The temperature of the body will immediately begin to
fall, due to its radiating heat to the walls of the enclosure; and
this will continue until the temperature of the body is the same as
that of the walls, when it becomes constant, and the body has
apparently ceased radiating. If the temperature of the walls
be now reduced, by immersing the vessel in a bath of lower tem-
perature, the temperature of the body will fall and it will again
be radiating heat. The body then, apparently, ceases to radiate
heat when its temperature has fallen to that of the walls of the
enclosure, and again begins to radiate heat when the walls are
lowered in temperature; and again ceases to radiate heat when its
temperature has fallen to that of the walls, and so on indefinitely.
If, initially, the temperature of the walls had been higher than
that of the body, heat would have been radiated from the walls
to the body.
Now, according to the theory of exchanges, the body does not
cease to radiate when its temperature has fallen to that of the
walls; but, the body and the walls are continually radiating and
absorbing heat. That is, it is assumed that, when the body is
at a higher temperature than the walls, it is radiating heat more
rapidly than it is absorbing heat, when at a lower temperature
than the walls, it is gaining heat more rapidly by absorption than
it is losing heat by radiation, and when at the same temperature
the rates of radiating and absorbing heat are the same.
Without being committed to this theory, it will be interesting
to note certain conclusions which must necessarily follow from it.
74. Emissivity* Experiments on radiant heat show that some
bodies emit heat, other things being equal, more copiously than
others. It is also foimd that bodies which are good radiators
are also good absorbers. The capability which a body has for
emitting heat is called its emissivity.
Suppose now, that we have two bodies, placed in a space, imper-
vious to heat. Then, according to Provost's theory of exchanges,
they will both radiate and absorb heat, even though they be at
82 HEAT
the same temperature. If now, one of the bodies absorbs heat
more readily than it emits heat, its temperature will rise; this,
however, is contradictory to experience. If, on the other hand,
one of the bodies radiates heat more readily than it absorbs heat,
its temperature will fall; which again contradicts experience.
It therefore follows, if Provost's theory of exchanges holds, that
bodies have precisely the same capability for radiating heat that
they have for absorbing heat. This appears to be in concordance
with experiment.
75. Stefan's Fonmila. We know that Newton's law of cooling
|s very limited in its application; i.e., it does not hold when the
difference of temperature between the body under consideration
and the surroimding mediiun exceeds 15°C. to 20°C. In other
words, it is only an approximate statement. Dulong and Petit
performed a nmnber of classical experiments by means of which
they endeavored to determine the law of cooling. Their exper-
iments were, however, limited in range of temperature; since,
the maximmn temperature reached was only about 240^C. From
their experiments, they deduced the equation, for the quantity
of heat lost per unit time,
Q=mk\k^-1); (1)
where Q is the quantity of heat, t the temperature of the enclosure,
the difference in temperature between the enclosure and the
radiating body, both measured on the centigrade scale, and m a
constant, depending upon the substance and the nature of its
surface. For k the value of 1.0077 was found.
Stefan, from an examination of the results obtained by Dulong
and Petit, deduced a formula for the loss of heat by radiation; i.e.,
= fc(ri*-r2*); (2)
where k is constant, and Ti and T2 are the temperatures, as meas-
ured on the ideal gas thermometer, respectively, of the radiating
body and the enclosure.
PEOPAGATION OF HEAT 83
Equation (2) appears to give results, in accordance with exper-
iments, up to temperatures of about 1700°C. to 1800°C. It,
however, appears from subsequent experiments, that Stefan's
formula is not rigidly true; and consequently will require modi-
fication. But, for practical purposes, Stefan's formula may be
considered correct for the limits of temperature as stated.
Considerable research work is still being done in regard to
radiation at high temperatures; and whether, or not, a simple
expression will finally be found which will be true for all temper-
atures, is an open question.
76. Convection. Referring again to the incandescent lamp,
and considering the dissipation of heat from the surface of the
bulb, we find that part of the heat is absorbed by the atmosphere
surrounding the bulb, and the remainder is transferred by radia-
tion. Due to the absorption of heat, the gases in contact with
the bulb become heated and therefore change in density. This
change in density destroys the equilibrium, in regard to pressure;
and hence, currents are established, called convection currents,
tending to restore equilibrium. In this manner, heat is conveyed
from one portion of space to another by currents in the atmos-
phere; i.e., the particles in contact with the bulb become heated
and are replaced by particles at a lower temperature. These
particles, in turn, become heated and are replaced by other
particles; each particle carrying away a certain amount of heat.
Since, in general, the density of liquids changes with change of
temperature, it follows, that when a liquid is not of a uniform
temperature throughout, convection currents will be established;
and these will, of course, tend to bring about equilibrimn. Thus,
if a vessel containing a liquid, be heated at the bottom, the liquid
in contact with the heated surface becomes less dense, rises, and
is replaced by a portion of the liquid of higher density, which in
turn becomes heated, is replaced by a denser portion, and so forth.
77. Conduction. If a body, such as a inetal rod, be heated at
one end, then it is foimd that the temperature along the rod
84 HEAT
gradually rises; i.e., heat is transferred wiihont the displacement
of matter. Or, to put it in another way, heat is transferred from
particle to particle, in a manner such that the particles maintain
their relative positions. The propagation of heat through a solid
is called candtLction.
Assmne that we are dealing with a homogeneous body, bounded
by two parallel plane surfaces, indefinite in extent, and that one
surface is maintained at a temperature ti, and the other at some
lower temperature T2. Then, after a certain time, steady condi-
tions will be established. Consider now, the simplest case pos-
sible, viz, a prism of constant cross-sectional area, normal to the
two surfaces, and extending from one surface to the other. Now,
since the two surfaces of the body are indefinite in area, we are
justified in assuming that there is no lateral flow of heat; i.e.,
the heat flows through the prism in parallel stream lines, and
the quantity of heat absorbed by the surface at a temperature
Ti, for a given interval of time, will be precisely
equal in amount to the quantity of heat given
oflF by the other surface, at a temperature T2,
during the same interval of time. Or, in other
words, the flow of heat through the prism will
have become uniform; and the quantity of
heat passing any section, parallel to the two
surfaces, will be the same throughout. Then,
as a fundamental principle, verified by experi-
FiG. 7. ment, the temperature slope, or the rate of
fall of temperature €dong the prism, is con-
stant. Hence, if the distance between the two surfaces is
represented by s, the temperature slope is
r='^'; (3)
where r is the temperature slope, or the rate of fall of temperature.
The temperature at any point may be found as follows: Let, as
in Fig. 7, a be the surface at a temperature of ti; b the surface at
I
I
I
I
|r
I
I ^.__
I
I
I
I
I
I
PROPAGATION OF HEAT 85
a temperature of ^2, and a the distance between the two surfaces.
The rate of fall of temperature between the two surfaces is given
by equation (3); and the fall of temperature from the surface a
to the plane x, parallel to the two surfaces, is
X
T; = -(ti-T2);
o
hence, the temperature of the plane x, is
Tx = Tl (ti — T2) (4)
S
Theory indicates and experiment verifies that the quantity
of heat which is transferred by a prism, such as has just been
discussed, is proportional to the area of the exposed surfaces, to
the time, and to the temperature slope. Stated symbolically
QozAtr] (5)
where Q is the quantity of heat transferred, A the cross-sectional
area of the prism, t the time, and r the temperature slope. To
make statement (5) an equality, we must introduce a proportion-
ality factor; i.e.,
Q=KAtr;
from which
^=1 («)
K is the ratio of the quantity of heat, passing any section, to the
product of the area of the section, the time, and the temperature
slope at that section. This ratio is called the coefficient of con-
ductivity of the substance; and, of course, diflfers for different
substances.
From equation (6) it follows that the coefficient of conductivity
K, of a substance, is numerically equal to the quantity of heat which
flows across a section of unit area, in unit time, when the tempera-
ture slope is unity. In the c.g.s. system, and using the centi-
grade scale, the coefficient of conductivity of a substance is numer-
86 HEAT
ically equal to the quantity of heat, measured in gram calories,
which flows across a section 1 sq.cm. in area, in 1 second, when
the temperature slope at the section is l^C. per centimeter.
78. Flow of Heat along a Bar. If a bar be maintained at a
constant temperature at one end, and the remainder of the bar
be exposed to a space of lower temperature, which is also main-
tained constant, the fall of temperature along the bar will not be
the same as that of the prism previously discussed. For, since
the bar is at a higher temperature than the enclosure, it will con-
tinually give up heat to the surroundings by radiation and con-
vection. Eventually, heat will be supplied to every portion of
the bar, by conduction, as rapidly as it is dissipated by radiation
and convection. That is, the temperatures along the bar will
finally assmne steady values. But, as previously stated, the tem-
peratiu'e slope along the bar will not be constant. For since, when
a steady condition has been assumed by the bar, the quantity of
heat which passes any section, for a given interval of time, is neces-
sarily equal to the quantity of heal which is dissipated from the bar
beyond that section, for the same interval of tims, it follows that
the quantity of heat which passes a section of the bar becomes
less as the distance from the end, which is maintained at a con-
stant temperature by the application of heat, increases. Hence
since, other things being equal, the quantity of heat, which passes
any section of the bar, is directly proportional to the temperature
slope at that section, it follows that the temperature slope
decreases with increase of distance from the heated end.
79. Determination of Coefficient of Conductivity. Since, it
is impossible to realize in practice those ideal conditions which
were assumed in the discussion of the flow of heat between two
parallel walls having areas of indefinite extent, recourse must
be had to other methods. A bar maintained at a constant tem-
perature at one end, and having the remainder exposed to a space
of constant temperature, furnishes a convenient means for deter-
mining the coeflScient of conductivity.
PROPAGATION OF HEAT 87
To do this, we proceed as follows: After the bar has assumed
a constant condition throughout, its temperature is ascertained
at a niunber of definite points along it; this is most conveniently
done by means of a thermo couple, which is calibrated by com-
paring with a standard thermometer. The results are then plotted,
differences of temperature between the bar and its enclosure as
ordinates and distances along the bar as abscissas. The curve
passed through the points so found, shows the difference of tem-
perature between the bar and the enclosure, throughout the length
of the bar; and the slope of the tangent, drawn to any point of
this curve is numerically equal to the temperature slope at that
section. This gives us r for equation (6) ; and A of this equation,
viz, the area of the section, is determined directly from the dimen-
sions of the bar. It now remains to determine Q/t, i.e., the quan-
tity of heat which passes a section per unit time. To do this, a
second experiment is necessary. The bar is now heated imtil
its temperature is uniform throughout and sUghtly higher than the
highest temperature on the curve for the rate of fall of temperature
along the bar. The bar is then placed in the enclosure, under
precisely the same conditions as obtained when the curve for the
rate of fall of temperature was determined, and its temperature
noted at definite intervals of time. From the data so obtained,
a second curve is plotted, differences of temperature between
the bar and the enclosure as ordinates and times as abscissas.
The curve so obtained is the curve of cooling; and the slope of
the tangent, drawn to any point of this curve, is numerically equal
to the rate of change of temperature of the bar, with respect to
time, for the particular difference of temperature between the bar
and its enclosure at that time.
Let it now be desired to determine the quantity of heat, which
passes in a unit of time, some particular section of the bar, repre-
sented by the point a, on the curve A, of Fig. 8. Curve A is the
curve representing the temperatures along the bar, and curve
B, the curve of cooling. If now, that part of the bar to the right
88
HEAT
of a be divided into elements, such as oft, so short, that without
appreciable error, the fall of temperature along the element may
be considered constant, then the temperature of the element may
be taken as the mean of the two temperatures at the points a
and 6. If this mean temperature be then projected across to the
curve of cooling B, and at the point c, so f oimd, a tangent be drawn,
then the slope of this tangent is niunerically equal to the rate
of change of temperature with respect to time, for a difference
of temperature equal in amount to the difference between that of
the mean temperature of the element ab and its enclosure. If
Fig. 8.
now, we take the product of the thermal capacity of iJie element ab,
and the rate of change of temperature just found, we obtain q/t,
the quantity of heat lost, per unit of time, by the element ab at
the instant when its temperature is defined by the point c. But,
since the temperatures of the various parts of the element ab are
constant, the mean temperature is a constant, and differs con-
tinually from the temperatm-e of the enclosure by an amount pre-
cisely equal to the difference of temperature as found from the
curve of cooling for the instant when the temperature is repre-
sented by the point c. Therefore, the element ab is continuously
losing heat, at a constant rate, equal in amount to the quantity just
found from the curve of cooling for the temperature represented
PROPAGATION OF HEAT 89
by the point c. In a similar manner, the quantities of heat, escap-
ing per unit of time, from the various elements to the right of the
element ab, are found. Taking the sum of the quantities of heat
so found, for all the elements to the right of ab, the quantity of
heat Q/t, of equation (6), which passes the section a, in a unit of
time, is found. From ^hich, by substitution, K is found.
Experiment shows, that, in general, the conductivity of solids
decreases slightly with increase of temperature.
80. Conductivity in Non-isotropic Substances. If there be
a source of heat at a point in an isotropic substance, i.e., a
substance having like physical properties in all directions, then
other things being equal, heat will be propagated with equal
speeds in all directions; and the temperatures at equal distances
in all directions from the source of heat, at any instant, will be
foimd the same. Or, in other words, the source of heat will be
the center of spherical isothermal surfaces. On the other hand,
substances which are non-isotropic do not conduct heat with equal
speeds in all directions. As an example, the conductivity of Ice-
land spar is greatest in the direction of the axis of symmetry, and
equal in all] directions at right angles to this axis. It will be
remembered that the coefficient of expansion for Iceland spar is
also greatest in the direction of the axis of symmetry, and equal
in all directions at right angles to this axis.
81. Non-homogeneous Solids. Tyndall found, by experiment-
ing with cubes of wood, that the speed of propagation of heat
is greatest, in the direction of the fibers; i.e., parallel to the length
of the tree, and least, parallel to the annual layers. And in a
direction normal to both the fibers and the annual layers, i.e.,
radial to a section of a tree, a value was foimd for the conductivity
slightly greater than that parallel to the annual layers; but, con-
siderably less than that parallel to the fibers. Wood, however,
on the whole is a very poor conductor in comparison with metals.
It will be of interest here to note that the speed of propagation
of soimd through wood is different for the three directions; i.e.,
90 HEAT
the speed of propagation is greatest, parallel to the fibers,
least, parallel to the annual layers, and radial to a section of the
tree, it is somewhat greater than it is parallel to the annual layers,
but considerably less than that parallel to the fibers.
82. Conductivity of Liquids. The determination of the co-
efficient of conductivity of a liquid is attended by difficulties which
are not experienced when dealing with solids. For, in the case of
liquids, if we wish to determine the true conductivity, convection
currents must be avoided. It is, therefore, necessary to heat the
column of liquid from the top. It is impossible here, to consider
all the necessary precautions which must be taken to insure
accurate results. The principle involved, however, is precisely
the same as for solids. That is, to determine accurately the
temperature slope along the colimm and the quantity of heat
passing a given section for a definite interval of time.
83. Conductivity of Gases. The determination of the coef-
ficient of conductivity of a gas is still more difficult than is the
determination of the coefficient of conductivity of a liquid. For,
in a case of a gas, not only must convection currents be eliminated,
but radiation must also be taken into account. This makes it
extremely difficult to obtain even fairly accurate results.
As a matter of interest, the following coefficients of conductivity
for a few substances are given. They are all expressed in the c.g.s.
system with the gram calorie as the unit quantity of heat. That
is, the numbers in the table represent in each case, the quantity
of heat, in gram calories, which passes a section 1 sq.cm. in area,
in 1 second, when the temperature slope is l^C. per cm.
Silver 1.01 Glass 0.002
Copper 0.891 Firebrick 0.0017
Aluminum 0.344 Cork 0.0007
Zinc 0.265 Paraffine 0.0002
Iron 0.167 Water 0.0014
Mercury 0.0152 Ether 0.0003
Ice 0.0057 Hydrogen 0.0004
Granite 0.005 Air 0.000066
PROPAGATION OF HEAT 91
The student must always remember that the results given in
the tables for the coefficients of expansion and conductivity must
be taken as being only approximate. For, the physical properties
of a substance depend very largely upon its chemical purity; and,
furthermore, the properties any substance may manifest, will
depend very largely upon its physical history and composition.
This is especially true for alloys, such as brass, organic growths,
such as cork, and complex compositions and mixtures, such
as glass.
It is interesting to note that, for metals, the order is the same
for electrical conductivity as it is for thermal conductivity; i.e.,
good conductors of heat are also good conductors of electricity,
and vice versa. However, there is not, as was at one time sup-
posed, strict proportionality.
THERMODYNAMICS
CHAPTER VIII
FUNDAMENTAL PRINCIPLES
81. First Principle of Thermodynamics. The first principle
of thermodynamics is merely the application of the principle of
energy to the special case of mechanical work and heat; and may
be stated as follows: When heat is converted into work, or work into
heat, the ratio of the numbers representing the two quardities involved
is a constant. The foregoing statement is, of com^e, the result
of direct experiment.
86. Second Principle of Thermodynamics. The second pririr
ciple of thermodynamics is stated variously by different authors.
Indeed, in some cases, the statement is preceded by discussions
which involve almost the whole theory of heat. For our purposes,
however, the statement first enimciated by Clausius will suffice.
This statement is essentially as follows: Heat cannot pass from
a body of lower temperature to one of higher temperature urUhovt
the aid of some external agent. This statement, though not the
result of direct experiment, is in conformity with our conunon
experience. As an example, we know from experience that heat
passes by conduction and radiation from regions of higher tem-
perature to regions of lower temperature. To illustrate further,
^assume that we are dealing with two bodies A and B, and that the
temperature of the former is lower than that of the latter; then
03
94 THERMODYNAMICS
heat may be made to pass from A to B, by applying heat to A,
until its temperature is the same as that of B, bringing the two
bodies into contact, and by the further application of heat to A,
heat will pass from it to B. Heat may also be made to pass from
A to B, it work be first done on the former, such as compressing
it, until its temperature is the same as that of B; then by bringing
the two bodies in o contact and developing, by a further expendi-
ture of work, more heat in A, heat will pass from it to B. But,
until there is a tendency to liaise the temperature of A above that
of B, no heat will pass from the former to the latter. Assume, now,
a third body, C, imder compression and at the same temperature
as -A. By bringing the two bodies A and C into contact, and allow-
ing C to expand against the external pressure, thus performing
work, its temperature will fall and a certain quantity of heat will
flow from A into C. The body C may now be removed from A,
and compressed adiabatically until its temperature is equal to
that of B, and then by bringing C into contact with B, and by a
further expenditure of work on C, heat will flow from it to B.
At the end of this process, C may be removed from B and allowed
to expand adiabatically, and, if the various ranges have been
properly chosen, it will at the end of this cycle of operations be
in precisely the same condition as it was at the beginning. But,
A now contains less heat, and B contains more heat than it did
when the process began; and since C is in the same condition
as it was at the beginning, heat has been transferred from a body
of lower temperature to one of higher temperature by the aid
of an external agent.
86. Heat Motors. Heat Motors , or Heat Engines, are devices by
means of which energy in the form of heat, is converted into
energy, in the form of mechanical motion.
All heai motors consist of three parts; viz, a source of heai, a
working substance, and a refrigerator. Furthermore, all actual
heat motors act periodically; i.e., operate on cycles; and for each
cycle a certain quantity of heat is abstracted from the source by
IW^^Stt?f'£r4inMRfi' ^^^*"^ makiDg a general
kM'tCBii«IEtiBiiS7a|'&ive, as an illustration, a
'j|^'U,^i||/,**)'^]^g^Qonucal, method for the
'V V(»*Ht^t7l(8fS^6i^oir of boiling water
||rgf<^l^<*^#g|[Si[{^t 0°C., and the working
jeaJSit^'^^-fl^'a^*^*"^!^^ rod ab. One end of
gg^ converted into work,
ititti, etc., is rejected to the
M
l^f^d the other end f^ainst
on an axis through 0. A
engages one of the teeth
drum, having wound over
llppose now, that the lei^h
t to its coefficient of Unear
IB raised from 0°C. to 100°
J;the disk turns through an
^f ■;^^|^eg:2|^G|^^^:|:tooth. The resistance R,
'" '"^ i«SEil£3i|^«|^^Jci^a^ the pawl will engage the
96 THERMODYNAMICS
next tooth. If the rod be now surrounded by a bath of melting
ice, it will contract and engage the next tooth; the pawl, in the
meantime, holding the disk in position. This process may be
repeated indefinitely, until the resistance has been displaced
through any desired distance. The cyde is then as follows : The
rod at the temperature of melting ice is put into position, and sur-
roimded by a i>ath of boiling water at a temperature of 100**C.
In consequence of this elevation of temperature, the rod expands
and turns the disk through a certain angle and in this manner does
work in overcoming the resistance R. The heat taken from the
source consists of two parts: One part being consumed in elevating
the temperature of the rod, and is numerically equal to the prod-
uct of the mass of the rod, its thermal capacity per unit mass,
and the elevation of temperature. The other part consists of
the heat equivalent of the work done in displacing the resistance
R. The rod, now being disconnected and surrounded by melting
ice, gives up to the refrigerator, in cooling from 100°C. to O^C,
an amount of heat precisely equal to that absorbed in being
heated, without any external work being done, from 0°C. to 100**
C. Therefore, the dififerenoe between the heat taken from the
source and that given to the refrigerator is equivalent to the work
done in displacing the resistance R. Since this completes a cycle
it may be repeated indefinitely without any change in the relation
of the quantities involved.
If, now, we represent by Qi, the quantity of heat absorbed from
the source, during a cycle, and by Q2, the heat rejected to the refrig-
erator, then the external work done is
Tr=J(Qi-Q2); (1)
where W is the external work done, and J the mechanical equiv-
alent of heat. Since for every cycle the quantity of heat Qi
has forever disappeared from the source, and only the part Qi— Q2
has been converted into work, it follows that, with the contrivance
FUNDAMENTAL PRINCIPLES 97
just described, it is impossible to convert all the heat, taken from
the source, into external work.
88. Heat of Expansion. As explained in Arts. 46 and 47,
when a gas is heated and expands against an external pressure,
the heat required is practically equal to that required to elevate
the temperature of the gas, plus the heat equivalent of the exter-
nal work done; i.e., the external work done, expressed in heat units,
is practically equal to the heat of expansion. This, however, is
by no means the case when a metal rod is heated and expands
against an external pressure; for, in this ca3e, the ?ieat of expansion
consists of two parts, viz, the heat equivalent of the external
work done, and the heat required to expand the rod against its
own inherent forces. The former may be called the external heat
of expansion, and the latter the internal heat of expansion.
In the present state of our knowledge we are unable to assign
the proper relative values for the heat consumed in elevating the
temperature of a substance and the internal heat of expansion;
but; for most substances, the latter is a relatively large quantity.
Since, now, in the cycle discussed in Art. 87, the internal heat
of expansion is not recovered as work, but is rejected to the refriger-
ator, it follows that such a contrivance cannot use heat eco-
nomically.
89« Camot's Cycle. The first scientific discussion of a peri-
odically acting thermodynamic engine is that due to Sadi Camot,
published in 1824. In this discussion, ideal conditions are assumed ;
i.e., it is assumed that there are no losses due to radiation and
friction. In other words, it was Camot's object to show that
under certain given conditions, assmning ideal processes, a definite
fractional part of the heat taken from the source, by a periodically
acting engine, is converted into work; and that, for the given
conditions, this is the maximal amount of work that may be
realized. The following demonstration will make this clear.
Assume that we are dealing with any working substance whatso-
ever, confined in such a manner that it may be put into contact
pinr:'M^VS'H*W'<iSi^VB w'E^''^^i'f whose temperatures
^SLVOdCB^gPif || [t^^^cess, and that the conduction
pi|||P^i^t^»ltt(||y both the source and refrig-
I^MBTllp'jIo^SIB^^'Ciiiagr ideal conditions, it must be
^StB'ji^^'S'l^'flfWf^^*^^*^ ^"'^^ ^^^ adiabatic
^■&WlSIKl1l!iStSI8>on the p-ti (pressure-volume)
9>KtS^B?ML@^ volume of the working sub-
t?|SriBi(]Bil»«^|it is removed from the refrig-
iilWf^a^'0'Mp^!ltStwt4i''3'' ordinates and volumes by
is now insulated and com-
Lce of which its temperature
i^Qied until the temperature of
at of the source, and its con-
i:^^t:^lume, is represented by the
tbstance during this compres-
der the curve; i.e., the area
now put into contact with
isothemally, by any desired
_ Is pressure and volume, at the
|;|C||^^^h^^J*^t^A||^bythe point C. During this
- ;. . j^. . j^. .^. .^. .g. .^. ..-!.
FUNDAMENTAL PRINCIPLES 99
isothermal expansion a certain quantity of heat Qi, has been
abstracted from the source, and work has been done by the working
substance, represented by the area BCFK, The working sub-
stance is now again insulated and allowed to expand adidbaticoMy,
in consequence of which its temperature falls. This expansion
is continued until the temperature of the working substance has
fallen to that of the refrigerator, and the work done by it, during
this expansion, is represented by the area CDEF. The working
substance is now put into contact with the refrigerator and com-
pressed isotherrrMUy until its condition, as regards pressure and
volume, is again represented by the point A. During this isother-
mal compression, toork was done on the working substance repre-
sented by the area DEGA ; and, a quantity of heat Q2, was rejected
to the refrigerator.
Since now, the working substance, as regards pressure, volume,
and temperature, is in precisely the same condition as it was at
the beginning of the cycle, its irUrinsic energy is also the same;
it therefore follows, from the first principle of thermodynamics,
that the difference between the heat abstracted from the source
and that rejected to the refrigerator, expressed in mechanical
units, is equal to the net work done. By an inspection of Fig.
10 it is obvious that the net work done is represented by the area
ABCD; and, from equation (1), we have
Wi=J{Qi-Q2);
where Wi is the net work done.
But the heat, expressed in mechanical units, abstracted from
the source is
W2^JQi;
therefore, the ideal coefficient ofcorwersion^OT the maximal fractional
part of the heat, abstracted from the source, which in an ideal
process can be converted into work, is
^ — jQi — ~Qr ^^^
100 THERMODYNAMICS
The result, just found, has been deduced without making any
assumption in regard to th^ nature of the working substance;
it is therefore perfectly general.
Suppose that the working substance suffers a physical change
of state during the cycle; the foregoing demonstration still holds.
For, since the working substance is in precisely the same condition
as regards pressure, volume, and temperature at the end of the
cycle as it was at the beginning, it follows that whatever physical
changes of state have taken place during any part of the cycle,
changes of a like kind must have taken place in the reverse order
during some other part of the cycle; and hence, are balanced.
Therefore, the difference between the heat taken from the source
and that rejected to the refrigerator, expressed in mechanical
units, is equal to the external work done.
90. Since the relation, expressed in equation (2), was deduced
without considering the properties of the working substance, it
must be independent of those properties. There being, however,
no other quantities involved in the right-hand member of this
equation, excepting quantities of heat, and since these do not
depend upon the properties of the working substance, they must
be functions of the two temperatures. That is, the quantity of
heat taken from the source must be some function of the tempera-
ture of the source, and the quantity of heat rejected to the
refrigerator must be some function of the temperatxure of the
refrigerator. Just what values are to be assigned to these func-
tions must be determined for some specific case, which is con-
sistent with the demonstration.
91. Camot's Cycle a Reversible Process. The ideal cycle just
described is a reversible process. For, if the working substance,
at that part of its cycle when its condition, as regards pres-
sure and volmne, is represented by the point A, Fig. 10, and its
temperature is the 'same as that of the refrigerator, is put into
contact with the refrigerator and allowed to expand isoOiermaUy
to the point D, it will abstract from the refrigerator, a quantity
FUNDAMENTAL PRINCIPLES 101
of heat Q2, and do an amount of external work, represented by the
area ADEG. The working substance is then insulated and com-
pressed adiabatically , in consequence of which its temperature will
rise; let this be continued until its temperature is the same as
that of the source, and its condition, as regards pressure and vol-
ume, is represented by the point C, and an amount of work,
represented by the area FEDC, has been done on the working
substance. The working substance is now put into contact with
the source, and compressed isothermaMy until its condition, as
regards pressiu-e and volume, is represented by the point B,
a quantity of heat Qi being rejected to the source, and an aziiount
of work, represented by the areaiPCSX ,has been done on the work-
ing substance. The working substance is now insulated and
allowed to expand adiabaticaUy imtil its temperature has fallen
to that of the refrigerator; its pressure and volume being the same
as at the beginning, and the external work, represented by the area
BAGKy having been done by it. Taking the smn, we find that the
work done by the working substance is represented by the area
KBADE; and the work done on the working substance is repre-
sented by the area BKEDC. Finally, the net work done on the
working substance is represented by the area ADCB. But,
during this process, the quantity of heat Q2 has been taken from
the refrigerator, and the quantity of heat Qi has been transferred
to the source.
Since now, the working substance is in precisely the same
condition as regards temperature, pressure, and volume, as it
was initially, it follows that the difference between the heat
rejected to the source and that taken from the refrigerator,
expressed in mechanical imits, is equal to the net work done on
the working substance; i.e.,
TF=J(Qi-02).
92. It will now be shown that, for a given source and refrig-
erator, an engine operating on the Camot cycle, that is, a reversible
102 THERMODYNAMICS
engine, converts into work as large a fractional part of the heat
taken from the source as is possible under the assumed conditions.
To do this, we assmne that we have two engines A and B, operating
between the same source and refrigerator, the former acting
direct and driving the latter, which is nmning reversed. Let
Ha and Ha' be, respectively, the heat taken from the source and
that rejected to the refrigerator by the engine A during a given
interval of time; and likewise, let Ht and Hi/' be, respectively,
the heat transferred to the source and that abstracted from the
refrigerator by the engine B, during the same interval of time.
Assimie,now, that the engine A, which is nonreversible, can convert
a larger fractional part of the heat taken from the source into
work than could the engine B if it were running direct. We then
have
Ha'^Hg' Hb—Hb' ^rtv
Ha' ^ Hb' ^^^
Also, the work done hy the engine A must be equal to the work
done on the engine B, since the former is driving the latter; hence,
we have
W=J{Ha'-Ha")=J{m'-Hn (4)
From equation (4) it follows that the numerators of the inequality,
expressed by statement (3), are equal; hence
Ha K,Hb •
Also, from equation (4), we find
Hb — Ha ^^Hb — Ha ;
hence, since Hb is greater than Ha, Hb" must be greater than
Ha'. But, with a reversible engine, operating on the Camot
cycle between a certain source and refrigerator, the quantities
of heat involved are always the same for a definite amount of
work, no matter whether the engine is acting direct or reversed;
FUNDAMENTAL PRINCIPLES 103
it therefore follows that under the assumed conditions, the source
must be gaining heat, since the quantity of heat Ua taken from
it by the non-reversible engine is less than the quantity of heat
Uh rejected to it by the reversible engine. Likewise, since the
quantity of heat Ka' rejected to the refrigerator, by the non-
reversible engine, is less than J?/', that taken from it by the revers-
ible engine, it follows that the refrigerator is continually losing
heat. Hence, under the assumed conditions, we have a system
in which heat is being transferred from a body of lower temper-
ature to one of higher temperature, without the aid of an agent
external to the system. Since this, however, contradicts the
second principle of thermodynamics, we must conclude that the
assumption made, viz, that any engine can convert into work
a larger fractional part of the heat taken from the source than is
possible by means of a reversible engine, operating on a Camot
cycle between the same source and refrigerator, is in error. There-
fore, a reversible engine converts into work as large a fractional
part of the heat taken from the source as is possible under the
given conditions.
We may, however, consider this in another manner. Assume
that there is a third engine, operating between the same source
and refrigerator, abstracting heat from the source, and rejecting
heat to the refrigerator at a rate such that both the source and
refrigerator are maintained at a constant temperature. This
third engine may then be employed in doing external work, and
we have a system which is doing work without the expenditure
of energy. This, however, contradicts the principle of energy;
hence we must again conclude that the original assumption is in
error. Hence the conclusion that, /or any given conditions, no engine
can convert into work a larger fractional part of the heat taken from
the source than that converted into work by a reversible engine.
93. Reversible Engine as a Standard. It must be remembered
that all processes so far discussed, in this chapter, are ideal proc-
esses and cannot be realized in practice; in other words, since the
104 THERMODYNAMICS
cycles of an actual thermodynamic engine, are attended by friction
and radiation, they are necessarily irreversible. As a matter of
fact, as stated in Art. 27, all processes are irreversible. Reversible
processes are merely ideal; i.e., conceptions of perfect operations.
When we speak of the efficiency of a mechanical contrivance,
as being the fraction p, we simply mean that the output is the
fractional part p of the input. And the closer p approaches unity,
the nearer the machine is considered to be to perfection. But
here again, our standard is an ideal one; i.e., we are comparing
our actual machine with one that is ideally perfect. Since it
has been shown that a reversible engine, operating on the Camot
cycle, between a given source and refrigerator, converts into work
as large a fractional part of heat taken from the source as possibly
can be converted into work under the given conditions, we are
justified in taking this engine as a standard with which to compare
the performance of actual engines.
94. Camof s Cycle with a Perfect Gas as a Working Sub-
stance. Assume that we have confined in a cylinder, by means
of a frictionless piston, a perfect gas, and that there is a source
of heat at a temperature Ti, and a refrigerator at a temperature
T2. Let the condition of the gas, as regards pressure and volume,
be represented by the point Ay Fig. 11, when put into contact
with the source; pressures being represented by ordinates and
volmnes by abscissas. We will, furthermore, assume ideal
conditions; i.e., perfect conduction for the isothermal processes,
perfect insulation so that adiabatic processes may take place, and
no losses. Consider now, the four processes as follows :
(1) The cylinder containing the gas at the pressure pi, volume
vi, and temperature Ti, is put into contact with the source of
heat at temperature 7^i, and the gas is allowed to expand isother-
mdHy by any desired amount, say to the point B\ its pressure
now being p2 and volume V2- During this expansion work is
done on the piston by the gas, measured by the area imder the curve
AB, and a quantity of heat Q\ is taken from the source. It being
FUNDAMENTAL PRINCIPLES
105
assumed that the temperature of the source during the abstraction
of the heat Qi, remains constant; this may be brought about by
supplying heat to it at a proper rate.
(2) The cylinder is now removed from the source, is perfectly
insulated, and the gas is allowed to expand adiabatically, in con-
sequence of which its temperature falls, due to the fact that the
external work is done at the expense of the intrinsic energy of
the gas. This expansion is continued until the temperature of
the gas has fallen to T2, that of the refrigerator; in the meantime,
(P..V
V
FlQ. 11.
work has been done on the piston by the gas, measiu*ed by the area
under the curve BC. The pressure is now pz and the volume ^3.
(3) ThQ cylinder is now put into contact with the refrigerator
and the gas is compressed isothermcdly, until its pressure is p4,
and volume »4, as represented by the point D. During this com-
pression a quantity of heat Q2 is developed, and work is done by
the piston on the gas, measured by the area under the curve DC.
It being assmned that the temperature of the refrigerator, during
the absorption of the heat Q2, remains constant; this may be
brought about by abstracting heat from it at the proper rate.
(4) The cylinder is now removed from the refrigerator, is
perfectly insulated, and the gas is compressed adiabatically until
106 THERMODYNAMICS
its temperature is Ti, that of the source, and its pressure and vol-
ume are, respectively, pi and vi. During this compression work
was done, by the piston on the gas, measured by the area under the
curve AD. The gas being now in precisely the same condition
as it was initially, its intrinsic energy must also be the same.
Since, now, A and B are on the same isoth&rm, and likewise,
C and D are on the same isotherm, we have, from the character-
istic equation,
Pit'i=P2V2=-Bri, (5)
and
P3f3=P4V4=-Kr2 (6)
Also, since B and C are on the sam^ adiabatic, and likewise,
A and D are on the same adiabatic, we have
P2V2** = P3t'3^ (7)
and
pit;i»=p4t;4" (8)
From equation (5) we find
and
From equation (6) we find
and
Pi = -r-, (9)
P2 = ^ '. (10)
P3 = -— , (11)
B>T2 /-rt\
P4=— (12)
Substituting the value of p2 as given in equation (10), and that
of p3 as given in equation (11), in equation (7) we find
^^1 , RT2 „
— -V2''=' va'*;
V2 Vz
FUNDAMENTAL PRINCIPLES 107
from which
V2 \
1
Again, substituting in equation (8) the values of pi and p4>as given
by equations (9) and (12), we find
RTi „ RT2 „
Vl ^4
from which
(14)
From equations (13) and (14) it follows that
from which
V3 V2
(15)
va Vl
Equation (15) shows that the volume at C must be to the
volume at D, as the volume at JS is to the volume at A, so that
when the gas is compressed adiabatically from Z), it will come to
the point A.
Since we are dealing with a perfect gas, its intrinsic energy is
a function of the temperature only, and is, therefore, independent
of pressure and volume. Therefore, the work done by the gas in
going along the adiabatic from £ to C, is exactly equal to the
work done an the gas in going along the adiabatic from Z> to il.
Hence to obtain the net work done during the cycle, and the quan-
tities of heat involved, it is only necessary to consider the two
isothermal processes; viz, the heat abstracted from the source,
and the external work done, by the gas in going from A to B,
along the isotherm Ti, and the heat rejected to the refrigerator,
108 THERMODYNAMICS
m
and work done, on the gas, in going along the isotherm T2 from
CtoZ).
Since the temperature, during the isothermal expansion, from
A to £ is constant, it follows that the heat abstracted from the
source is directly proportional to the external work done. Hence
we have
Qi^A { pdv;
where A is the heat equivalent of a unit of work. But p, for
any part of this process is equal to RTi/v;- hence
dv
Jvi V
=ARTi\og^ (16)
By similar reasoning we find, that the heat rejected to the
refrigerator, during the isothermal compression, in going from
C to D, is
Q2=A I pdv.
But, for any part of this process p is equal to RT2/V; hence
Q2=ART2r-
=ART2log^ (17)
The net work done, measured in heat units is, by condition,
proportional to the difference between the heat taken from the
source and that rejected to the refrigerator; hence
iiTr=Qi-02=Ai2^riiog^-r2iog^);
FUNDAMENTAL PRINCIPLES 109
and the ideal coefficient of conversion, since Qi has forever dis-
appeared from the source, is
^ ^ Wrilog^-7'2log^^
«' ^Bniog?
Vl
from which, since by equation (15), v^hx—vz/v^^ we find
^ — qT" — rr ^^^^
Equation (18) shows that, for a perfect gas operating on a
Camot cycle, the xdM coefficient of conversion is the ratio of the
difference in temperature of source and refrigerator, to the tem-
perature of the source, as measured on the ideal gas thermometer.
Equation (18) is usually written in the following form:
S — XL ,^-.
•n — -g-; (19)
where S is the temperature of the source and R that of the refrig-
erator, both being measured by means of the ideal gas thermometer.
96. A little consideration will showthat the foregoing discussion,
and result obtained, is perfectly consistent in everyway with that
of the Camot cycle using any working substance. We are there-
fore justified {Art. 90) in assuming that even under ideal conditions
the maximum quantity of work that can be realized from an
engine working between a given source and refrigerator and ab-
sorbing the quantity of heat H from the source, is
W=Jh{^) (20)
Writing equation (19) in another form, we have
110 THERMODYNAMICS
from which it is obvious that, for y) to approach unity, R must
either approach zero, or 8 must approach infinity. Experience,
however, shows that it is not economical to attempt to maintain
the refrigerator at a temperature lower than that of the surround-
ings. Also, as the temperature of the source is increased, a point
is soon reached for which radiation and pressures become excess-
ive, and lubrication becomes difficult. It therefore follows,
with conditions such as obtain on the earth's surface, that even
a perfect engine can convert only a small fractional part of the
heat, taken from a source, into work.
96. Reversible Engine and Refrigeration. An engine operating
in a reverse order, i.e., one that is taking heat from a body of
lower temperature, transferring heat to a body of higher temper-
ature, and absorbing external work, constitutes a refrigerating
machine. Let, for any given time. Hi be the quantity of heat
transferred to a body of higher temperature, and H2 the quantity
of heat abstracted from a body of lower temperature, by a per-
fectly reversible engine; i.e., a perfect refrigerating machine.
We will then have the following relation:
H1—H2 __ S—R .^i\
Hi ~ S ^^^^
From equation (20) we have, for the amount of work that must
be done, to transfer the quantity of heat Hi, to the body of higher
temperature,
W==JHi^^ (22)
In general, however, in the case of refrigerating machines, we are
concerned principally with the work that must be done to bring
about a certain absorption from the body of lower temperature;
i.e., the amount of refrigeration. It is therefore advisable to
deduce an expression for the amount of work that must be done
in terms of H2, the quantity of heat taken from the body of lower
FUNDAMENTAL PRINCIPLES 111
temperature, instead of the quantity of heat Hi, rejected to the
body of higher temperature.
From equation (21) we find
H2 __R
Hi~P
from which
Hi=H2^ (23)
Substituting in equation (22) the value of Hi, as given by equation
(23), we find
W^JH^^; (24)
which gives the desired relation.
97. In Art. 95, it was stated that it is not economical to attempt
to maintain the temperature of the refrigerator lower than that
of the surrounding medium. We are now prepared to demon-
strate this mathematically.
Let Hi be the heat taken from the source, at a temperature
S, and let Ri be the temperature of the surroundings. If then the
temperature of the refrigerator be also Ri, the work that, under
perfect conditions, may be realized is
Wi=JHi^^ (25)
Assume now, that the refrigerator, by means of a reversible engine,
is maintained at some temperature R2, lower than Ri. The work
that can now be realized, by means of a perfect engine, is
W2=JHi^^ (26)
Subtracting equation (25) from equation (26), member by mem-
ber, we obtain, due to lowering the temperature of the refriger-
ator, for the gain in work,
W2-Wi^J^{Ri-R2) (27)
112 THERMODYNAMICS
To maintain the temperature 722 we must, by means of a revers-
ible engine^ abstract heat from the refrigerator at the same rate
that the direct engine is rejecting heat to it, and transfer heat to
the surroundings. The heat rejected by the direct engine is
H2^Hi-AW2=Hi'-Hi^~^=Hi^. . . (28)
The work that must be expended in transferring this quantity
of heat from the body of temperature i22, to the surroundings at
a temperatiu*e Ri, is
=J^{Ri-R2) (29)
By comparing equations (29) and (27), it is obvious that,
even under ideal conditions, the amount of work that must be
done by the reversible engine, to maintain the temperatiu^ of the
refrigerator, below that of the surroundings, is equal to the gain
in work by the direct engine, due to the lower temperature of
the refrigerator. It therefore follows that, even without consid-
ering losses, there can be nothing gained by attempting to have the
temperature of the refrigerator lower than that of the earth's sur-
face. As a matter of fact, if the' temperature of the refrigerator is
lower than that of the surroimdings, heat will continually pass
from the surroundings to the refrigerator, and the reversible
engine must do an amount of work greater than that given by
equation (29). Furthermore, due to imperfections of the engines,
the gain in work realized by the direct engine will be less than that
specified by equation (27), and the work that must be done on the
reversible engine will be greater than that specified by equation
(29) ; hence, there is a decided loss when the refrigerator is main-
tained at a temperature lower than that of the surrounding media.
FUNDAMENTAL PRINCIPLES
113
98. Thermod]niamic Scale of Temperatures. The thermo-
dynamic scale of temperatures^ which was first proposed by Lord
Kelvin, will be made clear by the following considerations. Assimie
a series of n perfect heat engines arranged in such a manner that
the refrigerator of the first engine is the source of the second
engine, the refrigerator of the second engine is the source of the
third engine, etc., and furthermore, that the beat rejected by any
engine is absorbed by the engine next lower in the scale. To show
that, if the difference in temperature between soiLrce and refrig-
erator for the various engines is the same, they are all doing the
same amount of work.
Let, as in Fig. 12, the two adiabatics, AB and CD, be cut by
the isotherms Ti, T2, Tz, etc., such that the temperature intervals
are all equal, and each equal to t; i.e.,
ri-r2=r2-r3=rn-rn-Hi=T.
The ideal coeflScients of conversion for the various engines, begin-
ning with the first, then are
ri' 2^' ¥3' ' "fn'
(30)
114 THERMODYNAMICS
If H is the quantity of heat absorbed by the first engine from its
source, during a given interval of time, then
n—ii — y Ziy-
is the heat rejected to its refrigerator, and absorbed by the second
engine, during the same interval of time. In a similar manner,
the quantity of heat supplied to the third engine is
„i2 _ „i^ i 2 — is __ Tr£3
Tl ^Tl T2 " T{
The quantities of heat supplied to the various engines, beginning
with the first, then are
"^7 "^jT* -"jTj • • • -"~y — } ^jT' ' ' ^ ^
Since, now, the work done by any engine of the series is equal
to the product of its ideal coefficient of conversion and quantity
of heat, expressed in mechanical units, absorbed by it, it follows
from expressions (30) and (31), that all the engines are doing the
sameamount of work; i.e.,
is the work done by each engine of the series.
The results just deduced, being independent of the properties
of any substance, a thermodynamic scale of temperature may be
established in the following manner: Assume a series of n heat
engines, working between a given source of temperature Ti, and
refrigerator of temperature Tn+u ^^ such a manner that the n
engines are all doing the same amount of work, and each engine
is absorbing the heat rejected by the engine next higher on the
scale. If we then designate the difference of temperature between
FUNDAMENTAL PRINCIPLES 115
the source and refrigerator of any one of these ideal engines, as
a unit of temperature, we will have a scale of temperatures inde-
pendent of any substance, and depending only upon the perform-
ance of a perfect engine. But, from the discussion just given,
we found that by assuming the temperature intervals, as measured
on the ideal gas thermometer, equal, the engines were all doing
the same amount of work; hence, the thermodynamic scale is
identical with that of an ideal gas thermometer; and differs but
slightly, for temperatures not exceeding 500°C.; from those as
f oimd by means of the ordinary gas thermometer.
CHAPTER IX
STEAM AND STEAM ENGINES
m
99. The proper design of a heat engine presupposes, on the
part of the designer, a knowledge of the construction of mechan-
ical contrivances; i.e., how to construct a machine which shall
withstand the stresses imposed upon it in the performance of
its duties, with the lowest cost. The expression, lowest cost, must
not be interpreted as meaning lowest first cost; but it must be
understood to mean that the interest^on the capital invested,
for both machinery and ground rent, plus depreciation, plus
cost of power lost, must be a minimum. This part of the subject
comes under the heading of machine design; and, properly speak-
ing, has nothing to do, except in so far cw fuel economy is affected
by the design, with the subject of thermodynamics. But, a thorough
knowledge of the characteristics of the working substance and the
changes it undergoes, during its various stages, is fully as impor-
tant, if not more so, in the designing of an engine, as is a knowledge
of machine design. It is for this reason, since steam is so widely
used as a working substance, that so much research work has been
done, to accurately determine its characteristics.
100. Steam Operating on Camot's Cycle. Assume that we
are dealing with a unit mass of water, at a temperature T2, which
corresponds to that of the refrigerator, and let its condition,
as regards pressure and volume, be represented by the point D
of Fig. 13. The water is compressed adiabatically until its tem-
perature is Ti, that of the source, and its condition, as regards
pressure and volume, is represented by the point A. If the water
116
STEAM AND STEAM ENGINES
117
is now placed into contact with the source, and the pressure is
maintained constant, vaporization will take place. Assume this
to be continued until all the water has been converted into satu-
rated steam, whose condition, as regards pressure and volume,
is represented by the point B, The steam is now allowed to expand
adiabatically imtil its temperature has fallen to T2, that of the
refrigerator; its pressure and volume being now represented by
the point C During this adiabatic expansion a certain amount
of condensation, which will be discussed later, has taken place.
The mixture of steam and water is now put into contact with the
Ar
,Tl
B
D^
^ Arf
V
Fig. 13.
refrigerator and compressed isothermally until complete conden-
sation has taken place, and its condition, as regards pressure and
volume, is again represented by the point D, Since, now, the
condition of the working substance, as regards temperature,
pressure, and volume, is precisely the same as it was initially,
its intrinsic energy is also the same. Therefore, the net work
done during the cycle is measured by the area DABC. Further-
more, since the process is ideally reversible, the ideal coefficient
of conversion is
the same as previously deduced for any working substance.
118
THERMODYNAMICS
101. Relation of Temperature and Density of Saturated Steam.
It is frequently of prime importance to know the density of satu-
rated steam for a given temperature; and it being difficult to
determine this relation by direct experiment, it will be shown how
it is found from the relation of pressure and temperature of a
saturated vapor, this being easily determined by direct experiment.
To show how to deterinine the relation of temperature and density
of the saturated vapor of a substance, it will be assumed that we
are dealing with a unit mass operating on a Camot cycle, as just
described, and an indefinitely small difference of temperature,
AT J between source and refrigerator. This is represented dia-
gramjnatically in Fig. 14, where T+dTis the temperature of the
A
T+^T
^
B
V
Fig. 14.
source, and T the temperature of the refrigerator. This being
a reversible process, the ideal coefficient of conversion is
TQ =
AT
T+AT'
which, in the limit, becomes
TQ =
dT
T'
If the quantity of heat taken from the source, in going from A
to By is Q, then the work done is
W^Jd
dT
(1)
STEAM AND STEAM ENGINES
119
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OF
TEMPERATURE AND PRESSURE
OF
SATURATED STEAM
I
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200
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176
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§126
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] 20 THERMODYNAMICS
The work done during the cycle may also be expressed in terms
of the initial and final volimies and the change in pressure dp,
corresponding to the change in temperature dT. That is, if a
is the volume of unit mass of the liquid, and 8 the volume of unit
mass of satiu-ated vapor, then the work done, during the cycle,
is
W={8-a)dp (2)
Now the right-hand members of equations (1) and (2) must be
equal; since they are expressions for the same amount of work,
hence
dT
from which
and
JQ^dT
-^^'4-% (3)
The quantity Q, in equation (3), represents the quantity of
heat required to convert unit mass of the liquid into a saturated
vapor at the temperature T, and may be replaced by r, the heat of
vaporization; hence, equation (3) becomes
In equation (4), a, the voliune of unit mass of the liquid, for
the temperature T, is readily found by experiment; and likewise
r, the heat of vaporization. dT/dp is found from the curve giving
the relation of temperature and pressure of the saturated vapor.
Hence, since J, the mechanical equivalent of heat, is known,
8 is determinate; and the reciprocal of this gives the density of
the saturated vapor.
STEAM AND STEAM ENGINES 121
As a matter of interest, the curve showing the relation of
temperature and pressure, for saturated steam, is given on
page 119.
102. Perfect Steam Engine and Boiler. In the previous dis-
cussions it has been assumed that all of the heat is taken in at
the highest temperature. This, however, is by no means the case,
even under perfect conditions, with a steam engine and boiler.
For the present, we will confine ourselves to the operation
of a reciprocating engine, which has supplied to it saturated steam
from a boiler. The reciprocating engine consists essentially of
the following parts: A source of heat, the boiler y where steam is
generated under a constant pressure, and hence, at a constant
temperature, a cylinder and piston, and a refrigerator, or condenser,
at constant temperature, by means of which the steam, after
expanding and doing work against the piston, is converted into
water and returned to the boiler. The cycle of operations is
as follows: The piston P is at the position as represented in the
diagram. Fig. 15, and the condition of the steam, as regards pres-
sure and volume, is represented by the point A, the point of
admission. That is, at this point, the valve in the pipe connect-
ing the boiler with the cylinder is opened, and steam is freely
admitted. The piston advances to the point B, while vapori-
zation takes place at the temperature Ti. To simplify matters,
we will assume that we are dealing with unit mass of water and
that complete evaporation has taken place when the volume is
represented by the point B. The quantity of heat then, taken
from the boiler, is ri, the heat of vaporization at the temperature
Ti. The line of admission, AB, is a straight line and parallel to
the axis of volumes, since vaporization has taken place at constant
temperature; and hence, at constant pressure. The external
work done, during this advance of the piston, is measured by the
BiieskABFE. The point B is the point of cut-off; i.e., the admission-
valve is closed, and the steam is allowed to expand adiabatically
until its temperature has. fallen to T2, that of the condenser.
\
122
THERMODYNAMICS
In the meantime, the external work, measured by the area BCGF^
has been done. The exhatLst-valve now opens, and the steam
remaining in the cylinder, is compressed isothermally, in contact
with the condenser, until complete condensation has taken place,
and the work represented by the area DCGE, has been done by
the piston. The condensed steam, at the temperature T2, is
returned to the boiler and heated from the temperature T2 to
that of Ti, thus completing the cycle. The net work done during
A Ti B
JG
Fig. 15.
this cycle is evidently measured by the area ABCD, and is neces-
sarily less, as will now be shown, dvs to not taking in all the heat at
the maximum temperature^ than that which could be realized by
a Camot cycle.
The maximum amount of work that could be realized from
an engine taking in an elementary quantity of heat dQ, at the
temperature T, between the temperatures T2 and Ti, operating
on a Camot cycle, and rejecting heat to the condenser at the tem-
perature T21 is
T-r2
dW^JdQ
T
(5)
STEAM AND STEAM ENGINES 123
But, dQ is equal to cdT; where c is the thermal capacity and dT
the change in temperature. And, since we are dealing with imit
mass of water, dQ is practically equal to dT; since for water c
is almost constant and equal to unity. Therefore, equation (5)
may be written
dW^J^^dT; (6)
from which we obtain, for the total work that could be realized,
under ideal conditions, from the heat required to elevate the tem-
perature of xmit mass of water from T2toTi,
W'=Jl ^-J^dT
=j(ri-r2-r2iog^j).. . . (?)
The work that, under ideal conditions, could be realized from the
heat taken from the source during vaporization, since this is
absorbed at constant temperature, is
v/_ r.- T1 — T2
W^'^Jn^^; (8)
where ri is the heat of vaporization at the temperature Ti,
Adding equations (7) and (8), we obtain for the total work
that may be realized, for the given conditions,
W, = W' + W''=j(Ti-T2-T2log^+ri'^^^ .
(9)
Since the total heat, abstracted from the soiu'ce, expressed in
mechanical units, is
J(Ti-T2+ri),
124 THERMODYNAMICS
the maximum work which would have been realized, had the
operation been on a Carnot cycle, is
W2=J{Ti-T2+ri)^^^^ (10)
Dividing equation (9) by equation (10), we find
W2 Ti+n-Tz ^ '■^
If now, in equation (11),
7r^|rlog^>r2, (12)
then W1/W2 is less than unity. To prove that the expression,
given by the inequality (12), holds, we assume that T2, the tem-
perature of the condenser, is fixed, and that Ti is a variable, which
may be represented by T; remembering that T is always greater
than T2f and that both are positive. Expression (12) may then
be written
y'^lo5^=ftr2; (13)
where A: is a proportionality factor. From this, we find
log^=fc(l-f) (14)
Substituting, in equation (14), for T/T2, a new variable, x, we have
logx=fc(l-i);
and, differentiating with respect to x, we obtain
T
*=a; = -jr; (15)
STEAM AND STEAM ENGINES 125
from which, if TJT^ equals unity, fc equals unity; and, if TJT^
becomes greater than unity, M, must be greater than unity or
equation (15) cannot hold. But this means that the left-hand
member of equation (13) must be greater than T^'y and hence
W1/W2 is less than unity. It therefore follows that a steam
engine, which rejects condensed steam to a boiler cannot, even
under perfect conditions, convert into work as large a fractional
part of the heat taken from the source as can an engine operating
on a Camot cycle, between the same limits of temperature.
To illustrate the foregoing, we will deal with a concrete case;
i.e., assiune the temperature of the entering steam, and of the con-
denser, respectively, 356°F. and 140^F. This gives: Ti = 816,*
72 = 600, and ri=865. Substituting these values, in equation
(11), we find
QiA.QAR 816X600 , 816
W^ ^^^+^^^"816:^600^^^600 ^,^
W2^ 816+865-600 =^^-^ ^^^ "^^*'
giving a loss of about 9 per cent due to not taking in all the heat
at the maximum temperature.
103. Unresisted Adiabatic Expansion of Steam. If dry
saturated steam is allowed to expand adiabatically, from a chamber
of given pressure to one of lower pressm-e, without doing work,
the steam becomes superheated. This is due to the fact that,
when the steam enters the chamber of lower pressure, eddy cur-
rents are developed; and as they subside, the kinetic energy,
possessed by them, is converted into heat. Since the process is
adiabatic, and no external work is done, the total heat cantentf
i.e., the total quantity of heat contained by the steam, will be
the same at the end of the process as it was at the beginning.
But since, the total heat of steam decreases as the pressure is
* According to recenl; experiments, the zero for the thermod3maniic scale
is 491. 65 ^F. below the melting-point of ice; but, in general, 492 is sufficiently
accurate.
126 THERMODYNAMICS
decreased, and the final pressure is lower than the initial pressure,
the steam must become superheated.
If the steam is not initially dry, then it will become drier by
unresisted adiabatic expansion. Assume that we are dealing with
a unit mass of a mixture of steam and water under a pressure
pi, for which the heat of the water and the heat of vaporization
are, respectively, hi and ri. The total heat of the mixture, then is
H=hi+qiri; (16)
where gi is the dryness; i.e., the fractional part of the liquid which
is present as steam. After expansion, since the total heat content
remains the same, we have
H=h2+q2r2; (17)
where A2, q2, and r2 are, respectively, the heat of the liquid, the
dryness, and the heat of vaporization for the final pressure p2.
Equating the right-hand members of equations (16) and (17),
we obtain
hi+qiri = h2+q2r2 (18)
By priming is meant the percentage of moisture present; and
if this is low, the steam may become superheated by the unresisted
adiabatic expansion, and q2j in equation (18), becomes unity.
It will be shown later how, under certain conditions, advantage
may be taken of this, and the initial priming determined exper-
imentally.
104. Resisted Adiabatic Expansion. If steam, the initial
priming of which is low, expands adiabatically in such a manner
that external work is done, it will become wetter.
If in equation (9), Art. 102, it is assumed that complete
evaporation has not taken place before the adiabatic expansion
begins, then the work, expressed in heat units, yielded per cycle, is
W=Ti-'T2-T2\ogp+qiri'^p; . . (19)
STEAM AND STEAM ENGINES 127
where qi is the dryness. The total heat absorbed, in elevating
the temperature of the water from T2 to Ti, and evaporating it
to the dryness qi, is
Hi==Ti-T2+qiri (20)
And since, imder the assmned conditions, the difference between
the heat abstracted from the source and that converted into
work, must be equal to H', the heat rejected to the condenser,
we find, by subtracting equation (19) from equation (20),
ff' = (Zin^'+r2log^ (21)
But, the heat rejected to the condenser, after adiaoatic expansion
to the temperature T2, must be equal to the heat liberated during
condensation, i.e.,
H' = q^2; (22)
where 52 is the dryness and r2 the heat of vaporization corre-
sponding to the temperature T2' Equating the right-hand mem-
bers of equations (21) and (22), we obtain
q2r2 = qiri jT + ^2 log ^r ;
from which
«-5fe+'»«S) (»)
Equation (23) enables us to compute the dryness, during
resisted adiabatic expansion, provided the initial dryness be
known.
In the next chapter, the relation expressed in equation (23)
will be deduced by a much simpler and shorter method.
CHAPTER X
ENTROPY
106. It is obvious that a substance, in going from one isotherm
to another, always suffers the same definite change in temperature;
and furthermore, that this change in temperature is independent
of changes in pressure and volimie. That is, the change in tem-
perature in going from one isotherm to another is independent
of the path pursued during the change. A good analogue of this
is the change in potential a body undergoes in going from a sur-
face of potential Vi, to a surface of potential F2, the change in
potential, F2— Vi, being independent of the path pursued in
bringing about the change.
It will now be shown that, in going from one curve to another,
both curves representing reversible adiabatic processes, there is
some definite constant change. Equation (19) of Art. 48 specifies
for reversible adiabatic processes
C,dT+pdv=0; (1)
from which, by substituting for p its value as obtained from the
characteristic equation,
pv^RTy
and separating the variables, we find
C,f+R^=0 (2)
By integrating equation (2), between limits, we obtain
C.log|-+Blogf=0; (3)
i 1 Vl
128
ENTROPY 129
where Ti and vi are, respectively, the temperature and volume
before the change, and T and v, respectively, the temperature and
volume after the change. Since equation (3) is equal to zero,
no matter what the limits of integration, it follows that there is
something which does not change during a reversible adiabatic
process. Integrating equation (2) for the primitive, we find
C.logr+iilogt;=A;; (4)
where A; is a constant of integration. Equation (4) shows that
the fundamental differential equation for a perfect gas, yields upon
integration for a reversible adiabatic process a constant. But since
T and v, in equation (4), may have any values whatsoever, pro-
vided, always, they are so related that the process is adiabatic,
it follows that, no matter what the range, there is some function
which remains constant: which conclusion is the same as that drawn
from equation (3). Hence, since there is some function which
remains constant during a reversible adiabatic change, there must
be some definite constant change in going from one curve, repre-
senting a reversible adiabatic process, to another curve, represent-
ing a reversible adiabatic process.
In equation (4), the constant k evidently represents some
particular condition for the gas, which remains constant, during
an adiabatic process; and its value depends upon the unit of
measure and zero chosen. The condition of a gas, as expressed
by equation (4), was called by Clausius the entropy of the gas;
and, as just stated, the numerical value of the entropy depends
upon the units chosen and the arbitrary zero from which it is
measured.
Equation (4) may now be stated as follows: The entropy
of a substance during a reversible adiabatic change remains
constant.
106. Change of Entropy. If the left-hand member of equation
(1) is not equal to zero, i.e., heat is either added or abstracted
130 THERMODYNAMICS
while the gas changes in volume and temperature, the process is
no longer adiabatic, and the equation becomes
dQ=C,dT+pdv;
from which, by substituting for p its value as obtained from the
characteristic equation, we have
dQ = C.dT+RT~ (5)
V
Dividing equation (5) by T, we obtain
f-cf +4 m
If we represent the entropy of the gas by q), equation (4)
becomes
<f=CvlogT+R\ogv;
and, if the entropy is variable,
d9=C.f +fl^ (7)
The right-hand members of equations (6) and (7) being equal,
it follows that
d9=f ; (8)
i.e., for a reversible process ^ the change in entropy is numerically
equal to the ratio of the change in heat to the temperature at which
the change takes place; the temperature being measured on the
thermodynamic scale.
The foregoing may be illustrated by equation (18) of Art. 94,
which states that for a Camot cycle operating on a perfect gas,
Oi Ti '
from which
i-i <»)
ENTROPY
131
Equation (9) shows that the change in entropy in going from
the adiabatic AD, (Fig. 16) , to the adiabatic SC, is the same whether
the change takes place along the isotherm AB or DC] since the
ratio of change in heat to the temperature at which the change
takes place is the same in both cases. It is obvious that the same
ratio holds for any other isotherm cutting the two adiabatics
AD and BC. It is, however, not necessary that the change take
place along an isotherm. For, assume as depicted in Fig. 16,
the irregular path ej to be cut by the two adiabatics aa' and 66',
which differ by an indefinitely small interval. The change in
V
Fig. 16.
entropy, in going from g to A, along the irregular path e/, may be
resolved into the two component changes; i.e., the change in
entropy, in going along the isotherm gi, which is d<f=dQ/T;
where dQ is the change in heat and T the temperature at which
the change takes place. The other component ih, being adiabatic,
involves no change in entropy; hence, the change in entropy,
in going from gtoh,is
But since, as has just been shown, the change in entropy, in going
from one adiabatic to another is the same for all isotherms, it
follows, since gi is an isotherm, that the change in entropy in
132 THERMODYNAMICS
going from g to A is equal to the change in entropy in going from
a to b along the isotherm Ti, and also to the change in entropy in
going from a' to 6' along the isotherm r2. Similarly, it can be
shown that the change in entropy in going along the irregular
path efy between any two adiabatics, is equal to the change in
entropy in going between the same two adiabatics along either the
isotherm Ti or 72. It therefore follows that the change in entropy
in going from the adiabatic AD, to the adiabatic BC, is always
the same and is independent of the path by means of which the
change is brought about.
The foregoing demonstrations establish the fact that we are
justified in making the assumption that there is a constant definite
change in going from one reversible adiabatic to another; and,
this being the case, it follows that during a reversible adiabatic
process, some function, which has been termed entropy, must
remain constant. It also follows that, since for a Carnot cycle
Ti T2'
the source suffers a diminution of entropy during a cycle, which
is precisely equal in amount to the entropy gained by the refrig-
erator.
107. Universal Increment of Entropy. The conduction of
heat, such as discussed in Art. 77, is an irreversible process; and
if, during a given interval of time, a quantity of heat Q is abstracted
from a source at the temperature Ti, then, if steady conditions
have been assumed by the prism, an equal quantity of heat will
be rejected diu*ing the same interval of time, to some receiver at a
lower temperature, say 72. The loss in entropy, of the source,
is then
Q
9i = jr'
ENTROPY 133
and the gain in entropy of the receiver is
The gain in entropy of the system, since that of the prism is
michangedy is
^~'f'-T2~¥r T2Ti ^^°^
Since the right-hand member of equation (10) is positive, it fol-
lows that the entropy of the S3rstem, due to conduction, has
increased; and further, since the work which would have been
realized on a Camot cycle, for the quantity of heat Q, oper-
ating between the same temperature limits, is
we see that the work, expressed in heat units, which has been
irrevocably lost, due to the quantity of heat Q being transferred
by conduction from the temperature Ti to 72, is numerically
equal to the product of change in entropy and temperature of the
receiver.
We will now consider this in a wider sense. Assmne, first,
an engine working direct, which is thermodynamically perfect;
i.e., one which maintains, during its operation, the sum of the
entropies of source and refrigerator constant. If now, the mechan-
ism upon which the engine does work is perfect and capable at
any time of restoring all the energy imparted to it, then the process
is perfectly reversible. This, however, is never the case; since
all processes are attended by friction and a consequent develop-
ment of heat, which is imparted, by conduction and radiation,
to the surroimding bodies, there is necessarily an increment in
entropy. To put it still more broadly, since heat can be only
partially converted into work, and all energy, by friction, ohmic
134 THERMODYNAMICS
resistance, hysteresis, impact, etc., is finally degenerated into heat,
it would appear that the entropy of the Universe, such as we
know it, is tending toward a maximum. And the most gener-
alized definition we can give, is :* The change in entropy that a system
undergoes during a given irreversible process is a measure of the
irreversibility of the process. This is indicated, in a limited way,
by equation (10).
108. The concept of entropy has been here introduced, not
on account of its great scientific value, in the domain of theo-
retical physics, but rather because so many of the discussions
of practical thermodynamics are simplified so largely by its use.
For our purposes, the two most important statements are: The
change in erUropy during a reversible process, is numerically equal
to the ratio of change in heal to the temperature at which the change
takes place] and reversible adiabatic processes are also isoentropic.
109. Temperature Entropy Diagrams. From the equation
where 9 is the change in entropy, it follows, inmiediately, that
dQ = Tdr,
and, for a reversible isothermal process, we have
Q=rj^'d9 = r(92-9i) (11)
Appljdng equation (11) to a Camot cycle, we have for the heat
abstracted from the source, during isothermal expansion,
Qi=ri(92-9i); (12)
♦For a comprehensive discussion of entropy, see Planck's "Thermo-
dynamik," and also "Acht Vorlesnngen uber Theoretische Physik," by the
same author.
ENTROPY
135
and during isothermal compression, for the heat rejected to the
refrigerator,
02 = 72(92-91) (13)
From equations (12) and (13), we find
Q1-Q2 T1-T2
TQ =
Q:
Ti
(14)
The foregoing may, conveniently, be represented diagram-
matically, by plotting the r-9 (temperature-entropy) diagram;
using temperatures as ordinates and entropies as abscissas. Let,
Fig. 17.
in Pig. 17, the point A represent the condition of the working sub-
stance, as regards temperature and entropy, at the instant it
is put into contact with the source. Expansion now taking place
at constant temperature, together with the absorption of the
quantity of heat Qi from the source, the entropy increases by an
amount
92— 91 =
(15)
and, the condition of the working substance, as regards temperature
and entropy, is represented by the point B, Since, during this
change, the temperature is constant, the line representing the
change in entropy is a straight line parallel to the 9 axis. Further-
136 THERMODYNAMICS
more, since during the adiabatic expansion, the entropy of the
substance remains constant and only the temperature varies,
this change is represented by the line BC parallel to the T axis;
where the point C represents the condition of the substance, as
regards temperature and entropy, when put into contact with
the refrigerator at the temperature. 72. Compression now taking
place at constant temperature, together with the rejection of the
quantity of heat Q2, to the refrigerator, the entropy decreases by
an amoimt
92-91=^; (16)
and the condition of the working substance, as regards temper-
ature and entropy, is represented by the point D. This change
in entropy is represented by the line CD parallel to the 9 axis.
Finally, during adiabatic compression, the temperature rises from
72 to Ti; this change being represented by the line DA parallel
to the T axis.
From equations (15) and (16) we find
Qi-02 _ri-r2
Qi Ti '
as before, and the Camot cycle on the 7- 9 diagram is represented
by a rectangle; the heat abstracted from the source, dming a
cycle, being measured by the area
FAS^=ri(92-9i),
and the heat rejected to the refrigerator is measured by the area
FDCE= 72(92-91).
The difference between these two areas is a measure of the heat
converted into work; i.e., the area
ASCD = (ri-r2)(92-9i)
is a measure of the external work done.
CHAPTER XI
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS
110. In Art. 102, it was shown analytically that, even under
perfect conditions, a steam engine and boiler cannot convert into
work as large a fractional part of the heat taken from a som'ce as
can an engine operating on a Camot cycle. We will now show
this by means of the T- 9 diagram. As a matter of convenience
it will be assimied that we are dealing with a unit mass of water,
and, furthermore, that its thermal capacity is constant and equal
to unity, between the temperature of the condenser and boiler.
Under these conditions, the entropy, per unit mass of water, for
any temperature T, is
C^dT . T ,,,
where To is the temperature corresponding to the condition from
which the entropy is measured, and c equals unity. Hence, we
have a logarithmic curve instead of a straight line for one of the
sides of the T- 9 diagram. Let, as in Fig. 18, the point D represent
the condition of the water, as regards temperature and entropy,
when it is returned to the boiler. The logarithmic curve DA,
then represents the change in entropy with respect to change in
temperature as the water is heated from the temperature T2,
that of the condenser, to the temperature Ti, that of the boiler.
The water is now evaporated at the temperature Ti, and its change
in entropy is represented by the line AB. From the point B,
this point representing the condition of the water, as regards
temperature and entropy, for complete evaporation, the steam
137
138
THERMODYNAMICS
expands adiabaticallyi its entropy remaining constant, until the
temperature has fallen to T2, that of the condenser; its condition
being now represented by the point C. The steam is now com-
pressed isothermally, at the temperature T2, until complete con-
densation has taken place, and the initial condition, represented
by the point D, is reached.
During the cycle, just described, the net work done, or the
heat converted into work, is measured by the area ABCD; and
9
Fig. 18.
the heat abstracted from the source, is measured by the area
FDABE. Hence, the ideal coefficient of conversion is
t) =
Area ABCD
Area FDABE'
(2)
Had the process been a Camot cycle, the ideal coefficient of
conversion would be expressed by
^ Area DG BC
'^'Ares^FGBE
(3)
The value of iq as given by equation (2) is obviously less than that
given by equation (3); which is in agreement with the results
obtained in Art. 102.
APPLICATIONS OF TEMPERATURE-ENTKOPY DIAGRAMS 139
111. Change of Dryness during Adiabatic Changes. Equation
(23), of Art. 104, may be found in a very simple manner from the
fact that the entropy of a substance, during reversible adiabatic
changes, remains constant. The gain in entropy, for a unit mass
of water, in being heated from a temperature To to the tempera-
ture Ti is, by equation (1),
rp
9i = logjr; (4)
and the gain in entropy in evaporating to a dryness gi, at the
temperature Ti, is
?2 = giyr; (5)
where ri is the heat of vaporization corresponding to the tem-
perature Ti. Taking the sum of equations (4) and (5), we find,
for the total change in entropy
I <P=<Pi+92=log j^+gijT (6)
f
! But for reversible adiabatic changes the entropy remains constant;
, hence, if, after the condition expressed by equation (6) has been
attained, the temperature due to an adiabatic change, which may
be either expansion or compression, changes to T2, and the drjmess
changes to q2, we must have
log jr+5i2r=log jr+g2jr; .... (7)
where r2 is the heat of vaporization corresponding to the temper-
ature T2. From equation (7), we find
which is the same as equation (23) of Art. 104.
112. Dryness by Means of Temperature-Entropy Diagram.
The change in dryness which a mixture of water and steam, or
140
THERMODYNAMICS
any liquid and its vapor, undergoes during adiabatic changes,
provided the initial dryness be known, may readily be found by
means of the r-9 diagram. Let, in Fig. 19, the curve DA be
T
plotted, to proper scale, using various values of 9= log — as abscis-
To
sas and the corresponding values of 7 as ordinates; then this
curve represents the relation of temperature and entropy for
T,
B B
^\ —
Fig. 19.
unit mass of water between the temperatures To and Ti, If now,
for various points along the cxxrve DA, between To and Ti, hor-
izontal distances be measured off towards the right, each distance
being equal to r/T, where T is the temperature corresponding
to the point, and r the corresponding heat of vaporization, the
curve of saturation BC is found. By construction, then, at any
temperature T, the horizontal distance between the curves AD
and BC represents the increase in entropy due to the heat added
to bring about complete evaporation at that temperature. Thus,
for the temperature T, we have
(p=-=7wo;
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 141
TEMP. DE0REE8 PAHR.
1
fe
5
§
g
§
g
§
-
N
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*^
^
—
o
^
—
lo
""^
*>*,
^
■^
^
w;.-
kJ
w
£
"**
>.^
-
"^
■*-«
-
■^
■*«,
p
•
_J
■^
^
-
oo
n
.
g
00
i'5
m z ID ■"
> o z
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<
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z
30 r
gfe
•<
h^
1^
^^
—
■*■
^
—
bi
^
--
^
-
-r
'^
^-
:i
!?
3
u-
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oc
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"^
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/^
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il
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142 THERMODYNAMICS
where r is the heat of vaporization corresponding to the temper-
ature T, and q the increment in entropy.
Let, now, the initial drjmess, at the temperature Ti, be gi, such
that the entropy is represented by the point E, Since the quantity
of liquid evaporated, at a given temperature, is directly propor-
tional to the quantity of heat added, and the increment in entropy
is also directly proportional to the quantity of heat added, it
follows that the increment in entropy is directly proportional
to the amount of evaporation. Hence, the initial dryness is
AE
If, now, adiabatic expansion take place, the entropy remains
constant, and the vertical line EF represents the relation of tem-
perature and entropy. Hence, the dryness corresponding to the
temperature T is given by
5=^ (8)
^ mo
In a similar manner, provided always the initial dryness be
known, the dryness corresponding to any temperature during
an adiabatic change, may be found. And it makes no difference
whether we are dealing with an expansion or a compression.
It is obvious that if a T- 9 curve be plotted to a convenient
scale, for water, together with the corresponding saturation curve,
in a manner as has just been described, between such temperature
limits as are likely to occm: in practice, we may at once, from such
a sheet, provided always the initial dryness be known, determine
the amount of dryness at any temperature for adiabatic changes.
As a matter of convenience, such a sheet is given on page 141.
113. Zero Curve. Assume that various horizontal distances
between the curves AD and BC are all divided into the same
number of parts, and each part is the same fractional part of the
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 143
total distance. If the points, so found, are connected by smooth
curves, as shown in Fig. 20, then the dryness along any particular
curve is a constant. For, the horizontal distance between any
curve, such as mn, and the curve AD, no matter at what temperature
the distance be measured, is always the same fractional part of
the total increment in entropy, at that temperature, due to vapor-
ization; and therefore, represents the same fractional part of
vaporization. If, now, an adiabatic, such as ab, be drawn, it is
found to cut the curve mn; i.e., the curve mn passes to the right
of the adiabatic ab, as the expansion progresses, and shows that
Fig. 20.
the steam becomes wetter during adiabatic expansion and drier
during adiabatic compression. If, on the other hand, an adiabatic,
such as cdj be drawn, it is found that the curve of equal dryness
op, as the expansion progresses, passes to the left of it; and shows
that the vapor becomes drier during adiabatic expansion and wetter
during adiabatic compression. It is thus seen that if the vapor
be initially quite dry, it becomes wetter during adiabatic expansion,
and if the dryness is very low, it becomes drier when expanded
adiabatically.
If the adiabatic at any point becomes tangent to the curve of
constant dryness, then there is at that point, no change in dryness
during adiabatic changes. By finding a nimiber of points on the
various curves where the tangent is vertical, and joining these
144 THERMODYNAMICS
points by a smooth curve, then at any point to the right of this
curve the dryness is decreased by adiabatic expansion, and at
any point to the left of this curve, adiabatic expansion increases
the dryness. Such a curve is known as the **zero curve"; and
for temperatures such as are common in practice, does not lie
very far from 50 per cent dryness.
114. Loss of Work Due to Using Steam Non-ezpansively.
Assume, in the first place, that no expansion whatsoever is allowed; •
but that at the instant of cut-off, the steam is put into contact
with the refrigerator, and condensation takes place at constant
voliune. Then the work lost, due to using the steam non-expan-
sively, is represented by the area BCK, of Fig. 15, for the ideal
case discussed in Art. 102.
As a matter of fact, during the greater part of the eighteenth
century, steam was used in this manner; i.e., the steam was not
worked expansively, but immediately after cut-oflF, the steam
was condensed by a jet of water, either in the cylinder, or in an
adjoining condenser. The result, however, is the same whether
the steam be condensed by a jet of water, inmiediately after cut-off,
or allowed to escape to a space of lower pressure; for in either case,
there is the same gradual diminution of pressure in the cylinder.
The conditions which obtain, when steam is used non-expan-
sively, are best studied by the aid of the T-(f diagram. In
Fig. 21, DA is the T-^ curve for the heating of unit mass of
water, AB the curve for evaporation at the temperature Ti,
BC the saturation curve, BE the curve for adiabatic expansion,
BnF the curve of condensation at constant volume, and FD the
condensation curve at constant temperature. The condensation
curve BnF is determined as follows: For any temperature T, we
have to determine a point n such that
mn
Q= — ;
mp
where q is the dryness corresponding to the temperature T.
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 146
The volume of the steam in the cylmder, remaining sensibly
constant, since the volume of liquid present is practically negli-
gible, we have
««=«i; (9)
where a is the volume of unit mass of saturated steam at the tem-
perature r, and si the volume originally occupied at the temper-
ature T\, 8 and «i, are found from steam tables. From equation
(9) we find
?=
8
(10)
A
T.
B
L
T
n^x""'^
^k
1
D
f/
^
_.__El„Ac
T,
■•-— — *^^
Fig. 21.
But, as has been previously shown, a relation must subsist, such
that
«=
mp
(11)
hence, by combining equations (10) and (11), we find
mn = m/p — .
8
(12)
Finding a number of points in this manner, the curve BnF
is determined; and the loss of work, due to using the steam non
expansively, is obviously measured by the area BFE,
116. Loss of Work Due to Incomplete Expansion. If there
is a partial adiabatic expansion before exhaust or condensation,
146
THERMODYNAMICS
then the T-9 diagram takes the form as depicted in Fig. 22.
DA is the T-q) curve for the heating of miit mass of water, AB
the curve for complete evaporation, at the temi)erature Ti, BC
the saturation curve, BG the curve for adiabatic expansion, to
the temperature 7", GnF the curve of condensation at constant
volimie, and FD the curve of condensation at the temperature
3^2. To determine the curve of condensation at constant voliune,
we must find a point n for the temperature T, such that
tnn
Q= — ;
mp
(13)
A
T,
E
\
L-
T'
G
\
-
—I—
^y^
\.
y^—Z^ir^
1
r -^ ' \
1
/
^ I \
/
T,
/
-" ! \
/
/
. ^— _— _J A
D
F
b EC
Fig. 22.
where g is the dryness corresponding to that temperature. Now,
the volimie occupied at the point G, corresponding to the tempera-
ture r, is
gV;
where g' is the dryness at the temperature 2", and s' the volume
of unit mass of saturated vapor at 7". Also, for constant volume j
gs=gV;
(14)
where q is the dryness at the temperature T, and » the correspond-
ing volume for unit mass of saturated vapor. From equation
(14), we have
.«
3=vr
5
(15)
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 147
Finally, combining equations (13) and (15), we obtain
mn=q'—mp (16)
Finding a number of points in this manner, the curve OnF is
determined. If the steam is initially not dry, the curve GnF is
determined in precisely the same manner; but, the curve BG is
shifted toward the left by a fractional part of the length AB,
depending upon the amount of initial priming.
The work lost, due to incomplete expansion, is measured by
the area GFE; and, by an inspection of the figure, it becomes
obvious that the loss of work decreases very rapidly as the expan-
sion is increased. As an example, were the expansion continued
up to the point a, the loss of work, due to incomplete expansion,
would be measured by the small area abE.
The greater the amount of expansion, after cut-off, the longer,
necessarily, the stroke of the piston; but, the longer the stroke,
other things being equal, the higher the first cost of the engine,
and the greater the loss of work due to friction. Hence, there
must be a point beyond which it is uneconomical to carry the
expansion. Besides increasing the friction, there are still other
losses introduced, by carrying the expansion too far; these will be
considered later. Just how far to carry the expansion so as to give
the best economy is a problem far too complex to be solved theo-
retically. At best, theory can only serve as a guide, and the most
economical expansion must be determined experimentally. For
a simple engine, the point of cut-off may vary from about one-third
to one-sixth of the total stroke; dependmg upon whether the engine
is running non-condensing or condensing. But, it must always
be remembered that the ratio of cut-off to length of stroke depends
upon various conditions, which will be better understood after
we have dealt with the actual behavior of the steam in passing
through the cylinder.
148
THERMODYNAMICS
116. Gain of Work Due to Superheating. If the steam,
after being completely evaporated, be superheated, the ideal
coefficient of conversion is increased. But, this must not be under-
stood to mean the same proportional gain in work; for, lubrica-
tion and packing become more difficult as the temperature is
increased; and when the temperature becomes very high, radiation
becomes excessive.
Let, in Fig. 23, DA be the T-(f curve for the heating of unit
mass of water, AB the curve for evaporation, BE the curve for
superheating, EI the adiabatic expansion curve, and ID the
curve of condensation. The curve BE^ for superheating, is found
as follows: The thermal capacity of superheated steam, for
temperatures such as are found in practice, is approximately
constant; hence, we have
from which
9
T
(p=ciog jt;
(17)
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 149
where 9 is the change m entropy m going from B to E, T^ the
temperature to which the steam is superheated, and c the thermal
capacity per imit mass of superheated steam. The thermal
capacity per unit mass of superheated steam is, as has just been
stated, practically constant, and is approximately equal to 0.480.
With no superheating, and expansion along the adiabatic
BJf the ideal coefficient of conversion is,
,_ ABJD
^ "ABGKD'
and, with superheating and adiabatic expansion along £/, we
find, for the ideal coefficient of conversion,
„^ ABEID
^ ABEFKD'
By an inspection of the figure, it becomes obvious that r{'>t{.
If the steam be superheated to a temperature, such that the
adiabatic EF passes through the point C, then the steam will
be just saturated after it has been expanded to the temperat\ire
7^2. To find the amount of superheating that will bring about
this condition, it is only necessary to equate entropies, for the
points E and C. The change in entropy, in going from Z) to ^, is
and the change in entropy, in going from D to C, is
^ '¥2
But, in order that the adiabatic EF pass through the point C,
'p' must equal 9"; hence.
150 THERMODYNAMICS
from which
log7'.=j^(^^-^^-log^^)+logTi. . . . (18)
By means of equation (18), Tg is readily found
117. Double-acting Engine. Up to the present, we have been
considering matters as though the engine were only single-acting;
i.e., admission and exhaust take place only at one end of the
cylinder. In general, however, this is not the case. By having
proper valve arrangements, admission and expansion take place
in one end of the cylinder while release and exhaust take place
in the other end. And, the engine is double-acting; thus prac-
tically doubling the capacity of the cylinder and giving a more
uniform distribution of the work for a rotation of the fly-wheel.
Hence, a single-acting engine, to carry its load properly, requires
a fly-wheel of greater inertia than does a double-acting engine.
118. Condensing Engine. By a condensing engine is meant
an engine which exhausts to a receptacle of some kind, called a
condenser, where the steam is condensed at a comparatively low
temperature, and the pressure in the condenser is maintained
constant and lower than that of the atmosphere by means of a
vacuum pump. In good condensers, the pressure is as low as
the equivalent of one inch of mercury. By a non-condensing
engine is meant an engine which exhausts directly to the atmos-
phere at practically atmospheric pressure.
Experience shows that, in general, when the supplied steam
has a pressure of 100 lbs., or over, despite the fact that a certain
amount of power is consumed in operating the vacumn pump,
there is a decided gain in economy, when engines are operated
condensing. Hence, in general, condensing engines are employed.
CHAPTER XII
ELEMENTARY STEAM AND ENGINE TESTS
119. Before proceeding to discuss the actual behavior of
the steam as it passes through the cylinder of an engine, a brief
description will be given of the methods pursued in determining
the dryness of steam. It is obvious, from the discussions in the
preceding chapter, that one of the essentials in studying the per-
formance of a steam engine is a knowledge of the condition of
the supplied steam. But, aside from the temperature-entropy
diagram, if we wish to plot an adiabatic for steam on the p-v
diagram, we must know the initial dryness.
According to calculations by Zeuner, the equation
?w"=A:, (1)
where A; is a constant, may be used for adiabatic changes. But
the value of n depends upon the initial dryness. This value
of n is given by the empirical equation
w = 1.035+0.1g; (2)
where q is the initial dr3mess. Equation (2) holds for all values
between 70 per cent and 100 per cent dryness. Hence, if an
adiabatic for steam is to be plotted, that value of n must be
used in equation (1), which is found by means of equation (2), for
the given initial dryness.
120. Throttling Calorimeter. It was shown, in Art. 103,
that steam becomes drier, during unresisted adiabatic expansion;
and furthermore, under proper conditions, if the initial priming
be low, the steam may become superheated. Depending upon
151
152
THERMODYNAMICS
J
v\
0O.
this principle. Professor Peabody designed a calorimeter, by
means of which the dr3mess of steam, provided the priming is
low, may be determined. In Fig. 24, & represents the supply
pipe, and A a vessel into which the steam expands; the rate of
inflow being regulated by the valve Fi, and the outflow by the
valve 72. G\ is a gauge, indicating the pressure of the steam
in the supply pipe, G2 a second gauge,
indicating the pressure of ^the steam
in the vessel A, and m a thermometer,
indicating its temperature. The valves
Y\ and F2 are so regulated that the
pressure in the vessel A is always con-
m -\r siderably less than the pressure in the
supply pipe. The vessel A is either
well lagged with some non-conducting
material, or else highly polished, to
reduce radiation to a minimum. After
the flow of steam has continued for
some time, steady conditions will
obtain; and, if the priming of the
x| ^1 steam in the supply pipe is low, super-
Pjq 24. heating will take place in the vessel
A, and the thermometer m will register
a temperature higher than that corresponding to saturated
steam, imder a pressure as registered by the gauge G2. Let
Ti be the temperature of saturated steam corresponding to
the pressure indicated by the gauge G2, and T2 the tempera-
ture registered by the thermometer; then T2— ti is the amoimt
of superheating. The total heat of unit mass of steam then,
in the vessel il, is
H'=Ai+ri+c('C2-Ti);
(3)
where W is the total heat, Ai the heat of the liquid, corresponding
to the temperature ti, r\ the heat of vaporization for this tern-
ELEMENTARY STEAM AND ENGINE TESTS 153
perature, and c the thermal capacity per unit mass for superheated
steam. The total heat, per miit mass, for the steam in the supply
pipe, is
fl"=A+gr; (4)
where H" is the total heat, ft the heat of the liquid for the tem-
peratiu'e t, which is the temperature of saturated steam for the
pressure as registered by the gauge Gi, r the heat of vaporization
for this temperature, and q the dryness of the steam in the
supply pipe. For an adiabatic flow, however, the total heat
for the two conditions is the same; hence, the right-hand mem-
bers of equations (3) and (4) are equal, and we have
A+gr=fti+ri+c(T2-Ti);
from which
^ fti+ri+c(T2-Ti)~ft . .
ff= (5)
r
In order to obtain reliable results, by the method just described,
the two gauges Gi and (?2 must be accurately calibrated. On
the other hand, a slight error in the thermometer does not
appreciably alter the result; since the amount of superheating
is necessarily small, the quantity of heat involved is small in
comparison with the other quantities. But, (he thermometer mvM
he of sufficient accuracy so that we may be assured that there is
superheaiing.
The amount of moisture that may be removed by throttling
depends, of course, upon the difference between the pressure of
the steam in the supply pipe and the pressure in the chamber
into which it expands. If the pressure in the chamber is equal
to that of the atmos|phere, and the pressure in the supply pipe
is 100 lbs. per square inch, then the dryness must be about 96
per cent so that all the moisture may be removed by throttling.
If the pressure in the chamber be reduced by means of a con-
denser, a greater amount of moisture may be removed. If the
154 THERMODYNAMICS
initial pressure of the steam be 150 lbs. per square inch, and
the pressure in the chamber is atmospheric, then about 5 per
cent priming may be removed by throttling.
121. Condensing Calorimeter. The dryness of steam may
also be determined by condensation. There are various methods
which may be pursued; one is to have a vessel partially filled
with water at some low temperature, and passing steam into
this until some convenient rise in temperature has been attained.
The quantities of water and steam are determined directly by
weighing, the initial and final temperatures of the vessel are
read from a thermometer immersed in it, and the initial temper-
ature of the supplied steam is determined from the pressure,
as indicated by means of a gauge attached to the supply pipe.
Let M be the water equivalent of the vessel and contents,
m the mass of the condensed steam, and ti and T2, respectively,
the initial and final temperatures of the vessel and contents,
then if g is the initial dryness and t the corresponding temperature
of the supplied steam, we have
Af(T2— Ti)+mA2 = w{jr+mA; (6)
where A2 is the heat of the liquid corresponding to the temperature
T2, r and A, respectively, the heat of vaporization and the heat
of the liquid, corresponding to the temperature t. From equa-
tion (6) we find
Jf(T 2-Ti)-m(ft-ft2) ,yv
Equation (7) was deduced on the assumption that there is no
radiation during the progress of the experiment. A correction
for radiation may, however, be applied by taking a curve of
cooling for the vessel. Due to the fact, that the mass of the
condensed steam is determined by a difference in weighing, and
that this mass is necessarily small in comparison with the mass
of liquid initially contained in the vessel, a serious error may
be introduced by an inaccuracy in weighing.
ELEMENTARY STEAM AND ENGINE TESTS 155
A better method than the one just described, is that of
passing the steam through a spiral tube, contained in a condenser
through which there is maintained a continuous flow of water,
in a manner such that the steam is completely condensed and
reduced in temperature to that of the outflowing condensing
water. After steady conditions obtain, the mass of steam con-
densed, during a given interval of time, is determined by col-
lecting it in a suitable vessel; and in a similar manner, by
collecting in a separate vessel, the mass of water which passes
through the condenser, during the same interval of time, is
determined. Knowing the initial and final temperatures of the
condensing water, together with its mass, the mass of the con-
densed steam and its initial and final temperatures, then the
dryness of the steam is determinate. Let M and m, respectively,
be the mass of the condensing water and condensed steam for the
same interval of time, ti and T2, respectively, the temperature
of the condensing water for inflow and outflow, and t the initial
temperature of the supplied steam, then, since the temperature
of the outflow and that of the condensed steam is the safne, we
have
Af(T2--Ti)+m/i2=»w^+wiA; (8)
where h2 is the heat of the liquid corresponding to the temper-
ature T2, r and A, respectively, the heat of vaporization and the
heat of the liquid corresponding to the temperature t, and q
the dryness. From equation (8), we find
M(T2~Tl)-in(ft-/t2 ) .... (9)
mr
Equation (9) was deduced on the assmnption that there is no
radiation during the progress of the experiment. If the tem-
perature of the vessel differs materially from that of the siu*-
roimdings, corrections for radiation must be applied for this
difference. Radiation may, however, be completely eliminated
156 THERMODYNAMICS
by regulating the inflow such that the vessel and contents are
continuously at room temperature.
122. Separating Calorimeter. Professor Carpenter devised an
apparatus by means of which the moisture, present in the steam,
is removed mechanically. The steam is passed from the supply
pipe into a chamber, where it strikes against a convex surface,
surrounded by a wire mesh, through which the escaping steam
must pass. When the steam strikes the cup, the water present
is separated, passes through the mesh, and is collected in the
chamber; the dry satm^ated steam, meanwhile, passes into an
outer jacket, which surroimds the chamber, and escapes from
an orifice at the bottom, where it is condensed and collected.
The quantity of water collected in the chamber is read directly
from a glass gauge, which has been previously calibrated; and the
quantity of steam which passes through the calorimeter is deter-
mined by condensing and weighing.
In making a determination, the valve in the pipe supplying
steam to the chamber is opened, and when steady conditions
obtain, a reading is taken on the glass gauge, and simultaneously,
the exhaust pipe is passed into the condenser. When the oper-
ation has been continued for a suflScient interval of time, the
gauge is again read, and at the same instant, the exhaust pipe
is removed from the condenser. The mass of steam, passing
through the apparatus for the given interval of time, is found
directly by the difference in weight of the condenser for final
and initial conditions. And this mass compared with the sum
of the two masses, i.e., the mass of the condensed steam and the
mass of the water collected in the chamber, gives the dryness.
One of the inherent difficulties common to all m£thod8, in deter-
mining the dryness of steam^ lies in obtaining a sample which is
a fair average of the steam supplied to an engine.
123. Clearance. The volimie swept out by the piston of an
engine, during a stroke, is equal to the product of the area of
the piston and length of its stroke. The voliune between the
ELEMENTARY STEAM AND ENGINE TESTS 157
pLston and cylinder head at the end of the stroke, plus the
\roliime of the supply and exhaust-passages leading to the admis-
sion and exhaust-valves, is called the clearance. The clearance
then is, that part of the volume through which the piston does
not sweep, and is readily found by closing the valves and
determining the volmne of water required to fill the space when
the piston is at the end of its stroke. A convenient way of
expressing the clearance of an engine is by a ratio; i.e., the ratio
of the volume of the clearance, to the volume of piston displace-
ment plus volume of clearance. The ratio of the clearance
volume to the piston area gives the equivalent length of
clearance.
The clearance of different engines varies considerably, depend-
ing upon the size of engine and type of valves used; and, other
things being equal, the clearance for small engines is relatively
larger than it is for large ones. In practice, depending on the
type of engine, the clearance may vary from 2 per cent to 10 per
cent.
124. Cushion Steam and Cylinder Feed. The mass of steam
which remains in the clearance-space at the end of the exhaust-
stroke, depends upon the time of closing of the exhaust-valve.
Thus, if the exhaust-valve does not close xmtil the exhaust-
stroke has been completed, then the pressure of the steam,
remaining in the cylinder, is the same as that of the condenser,
and the mass of the steam is equal to the product of the clearance-
volume and the density of the steam. If, on the other hand,
the exhaust-valve closes before the exhaust-stroke has been com-
pleted, then the pressure of the steam, remaining in the cylinder
at the end of the exhaust-stroke, will be higher than that existing
in the condenser; hence, in this case, the mass of steam remaining
in the cylinder is greater than that for a later closing of the
exhaust-valve. The steam remainmg in the cylinder, at the end
of the exhaust-stroke, is called the cushion steam; and the steam
drawn from the boiler, per stroke, is called the cylinder feed.
' 158 . THERMODYNAMICS
During expansion, both quantities are present, whereas, during
compression, the cushion steam alone is present.
126. Wire Drawing. If the exhaust-valve closes late, the
pressure of the cushion steam is less than that of the steam in
the supply pipe, and a certain quantity of steam must pass
into the cylinder, during each stroke, before the maximum
pressure is reached. The entering steam, therefore, does not
do as great an amount of work on the piston as it would do if
the cushion steam had been compressed to the pressure of the
incoming steam; in other words, it is a case of imperfectly
resisted expansion. It is true that the incoming steam, if dry,
becomes superheated, and if partially wet becomes drier, due
to the partially unresisted expansion; but the pressure being
lower, the heat which is evolved when the eddy currents subside,
is applied at a lower temperature, and therefore, the imperfectly
resisted expansion constitutes a thermodynamic drop.
If the exhaust-valve closes at the proper time, then the
pressure of the cushion steam is equal to the pressure of the
incoming steam, and the thermodynamic drop, so f a(r as this
part of the action of the engine is concerned, is avoided. Fur-
thermore, the work which is done on the cushion steam, in
compressing it from the condenser pressure to that of the incom-
ing steam, is precisely equal to the work done by it, in expanding
between the same limits of pressure; hence, there is no loss of work
involved due to compression. There are, however, other unavoid-
able losses. The piston advances rapidly, calling for a large sup-
ply of steam, and the admission-valve does not open instantan-
eously, but requires a definite time interval. Hence, due to the
resistance offered to the flow of steam, by the supply passages
and valves, there is a certain amoimt of throttling, the sameas
when the cushion steam is at a pressure lower than that of the
supplied steam; causing the pressure in the cylinder, during
admission, to be less than that existing in the supply pip)e. And
furthermore, in general, due to throttling, the pressure in the
ELEMENTARY STEAM AND ENGINE TESTS 159
cylinder gradually decreases as the admission advances. The
result of these combined causes, due to which the pressure in the
cylinder during admission is lower than that of the supply pipe,
or boiler, is known as mre dravnngy and constitutes a thermo-
dynamic drop.
There is also a loss of work during the exhaust-stroke, due
to the fact that the exhaust-passages and valves offer a resistance
to the flow of steam, which makes the pressure in the cylinder,
during exhaust, always higher than that of the condenser.
126. The Indicator. One of the most important and, at the
same time, one of the most delicate pieces of apparatus used in
engine testing is the indicator. The indicator consists essentially
of two parts; the first part being a small piston P fitted accurately
into a cylinder and controlled by a helical spring S. The spring
may be either inside of the cylinder or, as shown diagrammatically
in Fig. 25, outside. The type of indicators having the spring
above the cylinder are more convenient; and furthermore, since
the springs, in this form, are not subjected to the same fluctua-
tions of temperature, the results obtained are more satisfactory.
The cylinder of the indicator, by means of a short supply pipe
containing a cock, is tapped onto the cylinder of the engine,
over the clearance space, in a manner such that the steam in
the engine cylinder exerts its full pressure against the piston
of the indicator throughout the entire cycle. If the indicator pis-
ton moves freely, i.e., without appreciable friction, and the spring
obeys Hooke's Law, then the movement of the piston will be
proportional to the fluctuations of the pressure in the cylinder
of the engine. To magnify the motion of the piston of the indi-
cator, the end of its piston rod is connected, by means of a system
of links, to a lever, in a manner such that a pencil point p, carried
by the end of this lever describes, between the limits of travel,
practically a right line. The springs are accurately calibrated
to a definite scale with respect to the motion of the pencil point.
Thus, if a spring is a 60 Jb, spring, it means that the pencil
160
THERMODYNAMICS
point moves over a distance of 1 inch for a change in pressure, on
the piston, of 60 lbs. per square inch; and a distance of 0.75 inches
for a change in pressure of 45 lbs. per square inch, etc.
The second part of the indicator consists of a drum JD,
controlled by a spring, upon which the indicator card is wrapped.
The drum has wrapped around its lower part a cord C, which in
m^
m
«a T> i^
^1
00
Fig. 25.
turn is connected by means of some mechanism, to the cross-
head of the engine, in a manner such that the angular displace-
ment of the drum is proportional to the linear displacement
of the piston of the engine. The linear displacement of the
surface of the drum, however, is less than that of the cross-head;
i.e., the motion is reduced by the mechanism through which the
ELEMENTARY STEAM AND ENGINE TESTS 161
cord, operating the drum, is connected to the cross-head. From
the foregoing, it is obvious that, if the drum is stationary, and
the cock in the supply pipe between the cylinder of the engine
and the cylinder of the indicator is open, the pencil point traces
a straight line on the indicator card. On the other hand, if the
stop-cock is closed and the drum is in motion, the pencil traces
a straight line at right angles to the former. This line is the
atmospheric line, since the stop-cock is so arranged that when
the steam is cut ofif from the indicator cylinder, a vent opens,
allowing free access of the atmosphere to the space below the piston
of the indicator. If, however, the stop-cock is open, and the
driun is moving in unison with the piston of the engine, the
position of the pencil point of the indicator, at any part of the
cycle, is a measure of the pressure and volume, of the working
substance, for that instant. Hence, during a cycle, the pencil
point traces out a diagram, which shows to a reduced scale, as
regards pressure and volume, the condition of the working sub-
stance, for every part of the cycle. The diagram so traced, is
the actual indicator diagram of the engine.
127. Indicator Diagram and Valve Adjustment. By means
of the indicator diagram, the behavior of the working substance
may be conveniently studied for the entire cycle; and further-
more, we are enabled by it to judge, whether or not, the valves
are properly adjusted, which is very important; since any faulty
valve adjustment may seriously affect the eflSciency of the engine.
Also, as will be shown in this chapter, by means of the indi-
cator diagram, we are enabled to determine the power delivered,
by the working substance, to the engine; hence, if the power
delivered by the engine be known, the eflSciency of the engine,
as a mechanical contrivance, is immediately determined.
Fig. 26 is a reproduction of an indicator diagram taken from
one end of the cylinder of a 40-H.P. engine, making 300 r.p.m.;
the engine working non-condensing; i.e., exhausting to the atmos-
phere. Fig. 27 is the indicator diagram for the same end of
162
THERMODYNAMICS
the cylinder when the engme was exhausting to a surface con-
denser; a partial vacuum being maintained by a pump.
AB is the admission line, BC is the expansion line, C being
the point where the exhaust-valve begins to open, and D the
JH
A
I
A
.V-
\
a
^^ E
~A.
I
»
G
Fig. 26.
—J
point where it is fully open; DE m the exhaust line, and at
the point E compression begins. Just how far the compression
will be carried before the admission-valve opens depends upon
the set of the valves. In the diagrams here shown, the admission-
FiG. 27.
valve opened at the point a. The various points being illy de-
fined is due to the time element involved in the opening and closing
of the valves. The line OH is the line of zero volume, and is
found by taking a distance, to the proj)er scale, to the left of
FAf representing the equivalent length of the clearance. The
ELEMENTARY STEAM AND ENGINE TESTS 163
line of zero pressure, or vacuum line OJy is found by measuring
down from the atmospheric line FG, a distance representing
the atmospheric pressure at the time the diagram was taken.
Finally, HI shows, as registered by the gauge, the steam pressure
in the supply pipe.
128. Comparison of Theoretical and Actual Curves. If it be
desired to compare the expansion or compression curve, with
an isotherm or adiabatic, a point on the curve is chosen, preferably
about the middle, and the theoretical curve is made to pass through
this point. As a matter of convenience, the following method
for plotting curves is here given. To plot the curve whose
equation is
we proceed as follows: In Fig. 28, OA is the line of zero volmne,
OB the line of zero pressure, and a a point on the curve. Lay
off the line OC^ making an angle g with' the line OA, and the
line ODy making an angle a with the line OB, such that
l+tanp = (l+tana)" (10)
Draw ad parallel to OB, and dh making an angle of 45° with OA.
Now draw af parallel to OA, and through /, Jq making an angle
of 45° with OB] then the point 6, which is the intersection of
the line W), parallel to OS, with the line gh, parallel to OA, is a
point on the curve. For, if we represent, for the point a, the pres-
sure and volume respectively, by pi and v\, and similarly for the
point 6, by p2 and V2, we have
Pi=P2+P2tan p=P2(l+tan P); . • . . (11)
and
vi+vi tan a=r2;
from which
t;i"(l+tana)"=t;2'* (12)
164
THERMODYNAMICS
Multiplying equations (11) and (12), member by member, we
obtain
PiVi'*(l+tana)^=P2i'2''(l+tanp); . . . (13)
but, by construction, as stated by equation (10),
(l+tana)" = l+tang;
hence, equation (13) reduces to
and 6 is a point on the curve. In a similar manner the points
c and i are found, etc.
Fig. 28.
The value of n to be used, if we are dealing with steam, is
found by means of Zeuner's equation, which is equation (2) of
Art. 119. And if we wish to plot an isotherm, n in equation (10)
is made unity, thus making the angles a and ^ equal.
In drawing the lines OD and OC, some convenient value for
a, say 15° to 20° is assumed, and the value of ^ is found by means
of equation (10).
In general, the expansion and compression curves, obtained
by means of an indicator, for heat motors and compressors
conform very closely to the equation
ELEMENTARY STEAM AND ENGINE TESTS 165
where k and n are constants for a particular pro^ss. The value
of 71; depending upon the nature of the working substance and
the condition of operation, may lie anywhere between unity and
1.4.
To find the value of n from the indicator diagram we may
proceed as follows: The pressures and volumes corresponding
to the points a and 6 (Fig. 28) are determined by means of a
scale, and then, from the equation,
a value for n is found. Similarly, values of n are found for a
number of points along the curve; and these values will, in general,
agree closely among themselves. And the mean of the values
so found, compared with the ratio of Cp to Cv, for the given
substance, is an indication of how closely the curve, under con-
sideration, approaches an adiabatic.
A method for finding the value of n from the indicator card,
which has been found to give satisfactory results and is less labor-
ious than the one just described, is as follows: Lay oflf the line
OC (Fig. 28), making an angle ^ with the line OA; then choose
a point on the curve, such as a, and draw the line ad parallel to
OB, and through the point d, a line making an angle of 45** with
the line OA, and cutting the line OC at the point h. Through
A, now, a second line is drawn, parallel to the line OB, which
cuts the curve at some point 6; two lines, now, parallel to the
line OA, one through a and the other through 6, are drawn, and
through g, where the line through 6 cuts the line OB, a line is
drawn making an angle of 45° with the line OB and cutting
the line, through a, at some point /. Now, this point / lies on
a line OD, making some angle a with the line OB, Proceeding in
this manner, a nmnber of points, for the line OD, are found
which will lie very nearly in a straight line. Drawing a mean
line through the points so found, the angle a is determined. And
by substituting for ^ and a, in equation (10), n is found.
166 THERMODYNAMICS
It will be noticed that in this case the angle a is determined
from the angle ^ and the curve mider consideration; whereas,
in the construction first ^ven in this article the curve is deter-
mined by means of n and the angles a and ^.
129. Behavior of Steam throughout the Cycle. When an
engine is first started, the cylinder walls are, of course, at a
temperature much lower than that of the steam, and conden-
sation takes place during admission, expansion, and exhaust.
After a time, however, permanent cyclic conditions will obtain;
i.e., regular periodical fluctuations will have been established,
and each cycle, so far as practical conditions permit, will
be an exact reproduction of the cycles preceding. When these
permanent cyclic fluctuations have been estabUshed the incoming
steam, during admission, comes into intimate contact with the
cylinder walls, which are at a lower temperature, due to the cooling
of the lower pressure exhaust steam which has been in contact,
just immediately preceding, and condensation takes place. It
is true, that, due to wire drawing, a certain amount of drying
takes place; but, imless the supplied steam has been super-
heated, considerable condensation will take place during admis-
sion, and will continue during part of the expansion-stroke;
and may, in some rare cases, continue throughout the whole of
the expansion-stroke. In general, however, during expansion,
some point is reached when the temperature of the steam falls
below that of the cylinder walk, and reevaporation takes place;
i.e., a certain quantity of heat is abstracted, from the steam,
by the cylinder walls, during the earlier part of the stroke, and
a certain quantity of heat is abstracted, from the cylinder walls,
by the steam, during the latter part of the stroke. Even if the
quantity of heat abstracted from the cylinder walls were equal
to the quantity of heat given up to them, which is never the
case, there would still be a thermodynamic loss; for the heat
abstracted from the walls is applied at a lower temperature
than that absorbed by the walls. The heat abstracted from
ELEMENTARY STEAM AND ENGINE TESTS
167
the cylinder walls, to bring about reevaporation, during exhaust,
is completely lost, since it is all rejected to the condenser.*
It is found, by experiment, that the exchange of heat between
metal surfaces and perfectly dry gases, is very small even for
considerable differences of temperature; hence, the conclusion
that, the rapid exchange of heat between the steam and the cylinder
walls, in a steam engine, is due to a film of conducting moisture
which collects on the surface of the cylinder walls.
130. Change of Dryness during Expansion. To determine
the dryness during the expansion-stroke, it is necessary to know
the cylinder feed and cushion steam. To determine the cyhnder
feed, the exhaust steam, for a given interval of time, is condensed
and weighed; and for the same interval of time, the number
of working strokes made by the engine, is determined. From
this, the mass of steam per stroke, i.e., the cylinder feed is
foxmd. The cushion steam is found directly from the indicator
diagram.
Let, in Fig. 29, ABCD be the actual indicator diagram; OE
and OF, respectively, the axes of zero volume and zero pressm-e.
FiQ. 29.
determined as described in Art. 127, and D the point where
the exhaust-valve has been completely closed and compression
begins. If the assmuption be now made that, at the point D,
the steam is saturated, no serious error is introduced; for, since
* For a comprehensive discussion of the influence of cylinder waUs, see
" Thermodynamics of the Steam Engine," by C. H. Peabody.
168 THERMODYNAMICS
the mass of the cushion steam is always small in comparison
with the total mass of steam present, during the expansion, a
small error made in determining the cushion steam, will not
appreciably aflfect the saturation curve. On the assumption
then, that at the point D the steam is saturated, the mass of
cushion steam is readily found, by means of steam tables; since
its volume and pressure are given by the diagram. Taking
the sum of the cylinder feed and cushion steam, we have the
total mass of steam and water present during expansion; and
from this, the saturation curve GH may be plotted. That is,
the curve GH gives the volumes, for the various pressures, the
steam would have occupied had it been completely saturated.
If then, at any pressure such as 01, the horizontal line IK be
drawn, the dryness for that pressure is at once found by the rela-
tion
IJ
131. Exchange of Heat, during Expansion, between Steam
and Cylinder Walls. If we plot, from the p-v diagram, a T- 9
diagram, which is easily done with the aid of steam tables, the
transfer of heat, during expansion, between the steam and cylinder
walls, is readily found.
In the r-9 diagram. Fig. 30, CD is the saturation curve,
AC is drawn at a temperature corresponding to the pressure
at cut-oflf, and the point B is so located that the dryness g, for
the point of cut-oflf, as found from the j>-v diagram (Fig. 29),
is given by
AB
Taking in this manner the dryness, for various pressiu-es, on
the p-v diagram, and transferring to the T- 9 diagram, the curve
of dryness Bnu is foimd. The curve ux is the curve of conden-
sation at constant volume, and is foimd as described in Art. 115.
ELEMENTARY STEAM AND ENGINE TESTS
169
The point n, where the vertical line pn becomes tangent to the
curve Bnu, is the point of mininiiini dr3niess, and is given by
9'=
tnn
mo'
It is obvious from the diagram that, during the expansion,
up to the point n, the steam is giving up heat to the cylinder
walls; and during the remainder of the expansion-stroke, heat
is abstracted by the steam, from the cylinder walls. Since the
area under the curve is a measure of the heat abstracted, or
rejected, it follows that, during the expansion, the heat given
Fig. 30.
to the cylinder walls is to that taken from them as the area
pnBq is to the area pniU,
As previously explained, the pressure in the cylinder during
admission, due to wire drawing, is less than the pressure in the
supply pipe; and as stated in Art. 125, superheating may occur.
In general, however, on account of initial priming, even if there
were no condensation during admission, due to the incoming
steam coming into contact with the cylinder walls of lower
temperature, there would still be present a certain amount of
moisture. Most authors assume, in discussing the exchange of
heat between the steam and cylinder walls, that the steam is
170 THERMODYNAMICS
dry at cut-oflF. This assumption is neither justifiable nor nec-
essary. No prediction can be made unless the dryness of the
supplied steam is known. If, however, the .dryness of the steam
in the supply pipe is known, together with its pressure and the
pressure in the cylinder, during admission, the dryness of the
steam in the cylinder, during admission, had there been no con-
densationj is readily computed. Let this hypothetical dryness be
represented on the T- (p diagram (Fig. 30) by
Aa
«' 'AC-
Since, however, the actual dryness at cut-off, as found from the
indicator diagram, is
AB
it follows that an amount of condensation, represented by the
change in entropy Ba, has taken place during admission. Hence,
the heat given up to the cylinder walls by the steam, during admis-
sion, is measured by the area Babq,
Heat is also given to the cylinder walls dining compression;
this, however, is not entirely lost. Since, due to this, the tem-
perature of the walls is raised, and the condensation during
admission, is partially reduced.
132. Steam Jackets. The fluctuations in temperature of the
cylinder walls, as described in Art. 129, are the more pronoimced
the lower the speed of the engine. In other words, the higher
the speed of the engine, the smaller the interval of time during
which exchanges can take place between the cylinder walls
and the steam, and as the speed becomes very high the exchange
becomes very small. There is, however, another element to be
considered, viz, the cooling of the cylinder, due to the fact
that it is always at a higher temperature than the surroimdings.
This loss of heat must continually be made up by the incoming
ELEMENTARY STEAM AND ENGINE TESTS 171
steam; and hence, increases the condensation. This loss of heat
is partially prevented by having the cylinder jacketed by some
non-conducting material. In some cases, a steam-jacket is used,
which is maintained full of live steam, taken directly from the
supply pipe; and therefore, the pressiu-e of the steam, in the
jacket is usually slightly higher than the pressure of the steam,
during admission, in the cylinder. There is, therefore, less
condensation, dining admission, than there would be were the
steam jacket absent; and reevaporation begins earlier. On the
other hand, the jacket increases the area of the exposed surface;
hence, a greater loss of heat, due to radiation. If complete
reevaporation takes place before the exhaust-valve opens, the
steam during the exhaust-stroke is dry, and very little heat is
absorbed by it from the steam in the jacket. The question
then is, whether the thermodynamic gain, obtained by applying
the heat at a higher temperature, to bring about reevaporation
at the earlier part of the stroke, is greater than the energy lost,
in the jacket steam, to bring about this reevaporation, plus the
greater radiation and heat imparted to the exhaust steam. This
question can be answered only by experiment. Experiments
performed, on slow and moderate-speed engines, appear to
indicate a decided gain in economy, by using a steam-jacket.
In a great many cases, however, such discrepant results have been
obtained, that it is extremely difficult to say imder just what
conditions steam jackets are beneficial.
133. Brake Power. The output of an engine of low power,
is most conveniently measured by a friction brake, which is a
device by means of which the power, developed by the engine,
is absorbed in overcoming the friction applied to the surface
of its fly-wheel; the force required to prevent rotation of the
brake, being measured by a balance.
The most common form assumed by the friction* brake is
depicted in Fig. 31. It consists of a number of wooden blocks
fastened by means of bolts, to steel bands, wrapping, approx-
172
THERMODYNAMICS
imately, two-thirds of the circumference of the fly-%heel. The
wing-nut w on the bolt b makes it possible to vary the pressure
to any desired value. The tie-rod t, going from the lower part
of the bolt b to the lever, is merely to give rigidity to the brake.
The rim of the fly-wheel is provided with flanges, so that water
may be contained in it, to absorb the heat developed by the
work done, in overcoming the friction.
Assmne, now, that the fly-wheel is rotating in the direction
as indicated by the arrow. Then, due to friction, the brake
will tend to rotate in the same direction; and to prevent this,
a certain force is applied to the lever, at the point p. This force
Fio. 31.
is most conveniently measured by a balance; which may be
either a spring balance or a beam balance. Let the fly-wheel be
making N r.p.m. (rotations per minute), the net weight registered
by the balance, to prevent rotation, be W lbs., and d be the
horizontal distance between the center of the shaft and point
of contact p. Then, since power, is numerically equal to the
product of angular velocity and torque, we have, employing
the minute as the unit of time,
P = 2T:NWd ft.-lbs. per min.;
and since one horse-power is the equivalent of doing work at
ELEMENTARY STEAM AND ENGINE TESTS 173
the rate of 33,000 ft.-lbs. per minute, we have, for the brake
horse-power,
^•^•^•"337000" ^^^^
In the case of very small units, the torque is frequently meas-
ured by wrapping a canvas belt aroimd the pulley, and applying
tensions to its two free ends. The tensions are then varied,
until the machine is loaded to the desired amount, and measured.
The torque is then found by taking the product of the difference
between the two tensions and the radius of pulley plus one-half
the thickness of the belt. In this case, the heat developed by the
work done, in overcoming the friction, is also absorbed by water
contained in the pulley.
When testing high-power machines, it is neither convenient
nor desirable to make friction tests. One method used is that of
connecting the engine under test to an electric generator, whose
efficiency is known, and by means of an ammeter and voltmeter,
or else by a wattmeter, determining its output. From the
efficiency of the generator and the power delivered by it, the power
delivered to it, by the engine, is readily foimd.
Another method for determining the power delivered by an
engine, is to make the shaft, through which the power is being
transmitted, take the place of a transmission dynamometer. This is
accomplished by determining the amount of twist, which a definite
length of the shaft experiences, when transmitting the given
power. Then, from the length and diameter of shaft, its modulus
of rigidity, and the angle of torsion, the torque is readily foimd.
134. Indicated Power. The power expended on the piston
of an engine, by the working substance, as found by means of the
indicator diagram, is called the indicated power. During admis-
sion and expansion, work is being done by the working substance
on the piston; and during exhaust and compression, work is being
done by the piston, on the working substance. Hence, the net
work done by the working substance, during a cycle, is measured
174 THERMODYNAMICS
by the area enclosed by the indicator diagram. If then, the area
of the indicator diagram be determined and divided by the length
of the stroke, reduced to the proper scale, the average ordinate
is foimd. The average ordinate, so found, multiplied by the scale
of the spring, used in taking the diagram, gives the mean effective
pressure. The area of the diagram is most conveniently foxmd
by means of a planimeter. There are certain types of planimeters,
which are specially designed for determining the mean effective
pressure from an indicator diagram. This type of planimeter is
very convenient, inasmuch as it is only necessary to set it to the
length of the diagram, employing a scale corresponding to the scale
of the spring, used in taking the indicator diagram, and following
the outline of the diagram with the tracing point of the instru-
ment. The mean effective pressure is then given directly by
the reading on the scale.
The mean effective pressure is the average pressure on the
piston, during admission and expansion, minus the average
pressure during exhaust and compression; hence, it is the effective
pressure, due to which external work is obtained. If the indica-
tor spring has been calibrated to lbs. per square inch, then the
mean effective pressure is also given in lbs. per square inch; and
the total effective pressure on the piston is numerically equal
to the product of the mean effective pressure and the area,
expressed in square inches, of the piston. If we represent by P,
the mean effective pressure, in lbs. per square inch, by A the
area of the piston, in square inches, by L the length of the stroke
in feet, and by N the number of cycles per minute, then the net
work done on the piston, per minute, is
W=PALN ft.-lbs.;
and the indicated horse-power is
^•'^'^- " 33;000 ^^^^
ELEMENTARY STEAM AND ENGINE TESTS 175
136. Mechanical Efficiency. The indicated power of an engine
is always greater than the power delivered by the engine, by an
amount which is equal to the power consumed in overcoming
the engine friction. The ratio of the brake horse-power, to the indi-
cated horse-poiDer gives the mechanical efficiency; i.e.,
^•»=m- (i«)
136. Thermal Efficiency. The thermal effi/Aency of an engine is
given by the ratio of the power delivered by the engine to the power
due to the heat taken from the source. As an example, assiune a
steam engine to be taking M poimds of steam per minute from
a boiler, the total heat of which, per poimd, is H, Let the heat
of the water in the condenser be A, which we will assume is returned
to the boiler without losses. Then the heat, expressed in mechan-
ical units, which is taken per minute from the boiler, is
,JM{H-h) ft.-lbs.;
and if W represents the number of ft.-lbs. of work delivered per
mmute by the engine, then the thermal efficiency is
W
We will now illustrate equation (17) by a numerical example.
Assume an engine making 300 r.p.m., doing work against a friction
brake whose arm is 5 ft., and which requires a force of 135 lbs.,
applied at its end, to prevent rotation. If the engine is consiuning
16 poimds of saturated steam per minute, under a pressure of 80
lbs., and returns the water without losses directly to the boiler, from
the condenser, where the pressure is 2 lbs., what is the thermal
efficiency?
Substituting, in equation (17), we find
^ 2xX300X 135X5 ^ ' .
^^ = 778X16(11 82^94:2) = ^'^ ^'' ^"^*'
176 THERMODYNAMICS
where 1182* is the total heat of steam under a pressure of 80
lbs., and 94.2 the heat of the liquid, corresponding to the temper-
ature of the steam, under 2 lbs. pressure.
137. Commercial Efficiency. The commercial efficiency of
an engine is given by the ratio of the power delivered by the engine
to the power which a perfect heat engine, working between the same
temperature limits, would deliver. Let the symbols have the same
significance as in Art 136, then the work, per minute, which a
perfect heat engine would deliver, is
JM{H-h)—^^ ft.-lbs.;
and the commercial efficiency is
W
^^= CO (18)
JM{H-h)'^~~
Substituting in equation (18), the numerical data given as
an illustration in the preceding article, we find
^ 2xX300Xl35X5
Ee = TK^ = «iy -U per cent.
778X16(1182-94.2)^
This is the proper method of comparison; i.e., comparing the
actual performance of the engine with an ideally perfect engine,
operating between the same temperature limits.
When an engine exhausts to the atmosphere there is, of course,
no heat returned to the boiler by means of the condensed steam,
and the heat h, in equations (17) and (18), is lost. It is, however,
not proper to charge this entire loss of heat against the engine;
since, by proper arrangements part of the heat at least, contained
by the liquid, can be returned to the boiler.
There are other methods for rating the performance of engines,
which are in certain cases, very convenient. One is, specifying
* Taken from Peabody's Steam Tables.
ELEMENTARY STEAM AND ENGINE TESTS
177
the number of pounds of steam per B.H.P. hour, consumed by
the engine. Another is, specifying the number of B.T.U. per
B.H.P. hxmry or the number of B.T.U. per K.W. hxmr of energy
delivered to the bus-bar. The latter is especially expressive;
giving, as it does, the rating of the power plant as a whole.
138. Rankine's Cycle. Another important comparison may
be made by the aid of Rankine's cycle, the indicator diagram of
which is shown in Fig. 32. This indicator diagram is based on
the assumption that the cylinder of the steam engine has no
clearance and is perfectly insulated. AB represents the admis-
sion at constant pressure pi, BC represents the adiabatic expan-
sion to the pressure p2, and CD represents the exhaust, at constant
pressure p2.
Assume now, that we are dealing with a unit mass of liquid,
whose specific volume is a, and that the dryness, during admission,
is q\. If the specific volume of the steam, at the pressure pi, is
«i, then the volume of the mixture, at the point of cut-oflf, is
t;i = gisi+(l— gi)a=gi(si — a)+a=qiiJLi+(j; .
(19)
where {jli is the increment in volume due to complete evaporation
at the pressure pi. Since the pressure, during admission, is con-
stant, the work done by the steam, on the piston, is
pit;i = pi(gi[J.i+a),
(20)
178 THERMODYNAMICS
The work done on the piston, by the steam, during the adiabatic
expansion, must be equal to the difference between the intrinsic
energy of the steam before and after expansion; i.e.,
£?i-£?2=J(Ai+gipi-A2-g2p2); .... (21)
where Ei, hi, and pi are, respectively, the intrinsic energy, the heat
of the liquid, and the heat of disgregation, corresponding to the
pressure pi, and E2, A2, and p2 are, respectively, the intrinsic
energy, the heat of the liquid, and the heat of disgregation, cor-
responding to the pressure p2. Or, to put it in another way, the
work done on the piston, by the steam, during the expansion, is
the difference in the heat contentj expressed in mechanical units,
before and after expansion. The work done on the piston, during
exhaust, is
-p2W2=-p2(g2iA2+cj); (22)
•
where 92 and (12 are, respectively, the dryness and increment in
volume, due to complete vaporization, at the pressure p2. Taking
the simi of the right-hand members of equations (20), (21), and
(22), we find, for the net work done during the cycle,
W=J{Apxqi\Li+hi''Ap2q2\>'2'-h2+qigi''q2?2)+(pi-V2)<J. (23)
The second term of the right-hand member of equation (23) is
very, small in comparison with the other term, and may be
neglected; hence, since from equation (55), of Art. 55,
r=Ap(i+p,
we find, by substituting in equation (23) the proper values,
W=J{qiri+hi-q2r2-h2) (24)
If pi and p2 are known, the values for hi, fe, ri, and r2 are found
directly from steam tables; and by knowing qi, the value of
q2 is readily found by means of the T- 9 diagram.
ELEMENTARY STEAM AND ENGINE TESTS 179
From equation (24) it is readily seen that the heat converted
into work, is the difference between the heat taken in, during
admission, and that rejected, during exhaust. This must neces-
sarily follow from the assumption that there are no losses in the
engine. But, it must be remembered that in the foregoing dis-
cussion, the engine is considered as being independent of the boiler.
Hence, by taking the ratio of the work actually performed by the
engine, to the work as given by equation (24), a result is obtained
which serves as a basis of comparison with other engines operating
under similar conditions.
To illustrate, we will take the same numerical values as given
in Art. 136, excepting that 90 per cent initial dryness will be
assumed, instead of complete saturation. The value of r corre-
sponding to 80 lbs. pressiu-e is 899.8, and that corresponding to
2 lbs. pressiu-e is 1021.9. The dryness, after adiabatic expansion,
at the pressiu^e of 2 lbs. is found, by means of the r-9 diagram,
to be approximately, 75.6 per cent, hi and A2 are given, respect-
ively, by 282.2 and 94.2. Substituting these values in equation
(24), we have
Tr=778Xl6(0.9X899.8+282.2-0.755Xl021.9-94.2)
=2,817,000 ft.-lbs. per minute.
Taking the ratio of the work delivered by the engine, to that which
would be realized by the Rankine cycle, we find, for the efficiency
of the engine,
„ 2xX300Xl35X5 .-^
^= — 2;8i7;ooo — =^^-^ p"' ''^*-
CHAPTER XIII
CX)MPOUND ENGINES
139. If a heat engine is to convert a large fractional part of
the heat taken from the source, into work, the temperature dif-
ference between source and refrigerator must also be large. Hence,
other things being equal, for a steam engine to operate econom-
ically, it is necessary to have a large range in temperature, or what
amounts to the same thing, a large range in pressure.
But, when steam under a pressure of 100 lbs., and upward, is
supplied to an engine, the fluctuations of temperature in the cylin-
der become large; and, consequently, the condensation becomes
excessive. To illustrate tiie fluctuations in temperature, assiune
an engine receiving steam imder a pressure of 100 lbs., and reject-
ing to a condenser under a pressure of 1 lb. From the steam
ciure, of Art. 101, the temperatures corresponding to these two
pressures are, respectively, 328°F. and 102®F.; i.e., a range of
about 226°F. Such a large range in temperature means a con-
siderable amount of condensation during the earlier part of the
stroke, and a consequent reevaporation during the later part
of the stroke; but, as previously explained, this constitutes a
thermodynamic drop; i.e., a wasteful applioation of heat. To
obviate this excessive condensation, when high-pressure steam
is used, the expansion is made to take place in two or more cylin-
ders; and the engine is said to be a muUiple'expansion, or compound
engine. When the expansion takes place in two cylinders, the
engine is said to be a dovbU-expansion engine, when in three
cylinders, a triple-expan^on engine, etc.
180
COMPOUND ENGINES
181
140. Double Expansion. We will consider jSrst the most
simple case possible; viz, cylinders without clearance, no losses
whatsoever, and a receiver, between the two cylinders, of such
volume that the pressure in it, throughout the cycle, is constant.
That is, the high-pressure cylinder receives steam from the boiler,
which, during admission and expansion, does work on the piston.
The steam is then rejected, at constant pressure, to the receiver;
the pressure in cylinder and receiver, during the exhaust, being
identical. During the same interval of time, that this is taking
place in the high-pressure cylinder, the low-pressure cylinder
receives an equal mass of steam, from the receiver, which in turn
does work, during admission and expansion, on the low-pressure
piston. The steam is then expelled, under constant pressure,
to a condenser in which a low pressure is maintained.
The indicator diagram, representing the foregoing is shown
by Fig. 33. The diagrams, for the two cylinders, are drawn to the
same scale and are superimposed. ABCD is the diagram for the
high-pressiu*e cylinder, and DCEF is that for the low-pressure
cylinder.
It will be noted that the combined diagram ABEF is precisely
the same as would have been obtained if the expansion, from the
initial volume, as represented by AB, to the final volume, as repre-
182 THERMODYNAMICS
sented by FE, had taken place in a single cylinder. That is,
so far as indicated power is concerned, provided there are no
losses, it is immaterial whether the expansion, from the initial
volmne to the final volume, takes place in one cylinder or a num-
ber of cylinders. There is, however, for the case just discussed,
due to the fact that the fluctuations in temperature have been
reduced, a decided thermod3niamic gain. There is also an impor-
tant mechanical advantage when, other things being equal, expan-
sion takes place in two or more cylinders. For, if the area taken
up by the piston rod be neglected, then the stress existing in the
rod, at any instant, is proportional to the difference in pressure on
the two sides of the piston. This difference is a maximum, while
admission is taking place at one end of the cylinder and exhaust
at the other. By referring to Fig. 33, it is seen that, for the same
ranges in pressure, the maximum difference in pressure for the
single-expansion engine is measured by AF; whereas, for the
double-expansion engine, the maximum differences in pressure for
the high and low-pressure cylinders are measured, respectively,
by AD and DF. It therefore follows, that in a compoimd engine,
the piston rods may be considerably reduced in cross-sectional
area, as compsu'ed with that of a single-expansion engine. And
since, for the same initial and final pressures the total work done
is the same in either case, it follows that the average thrust on the
cranks must be the same for the compoimd engine as it is for the
single^xpansion engine. But, in the case of the compound engine,
the thrust is more uniformly distributed throughout the cycle;
•
hence, less friction and a consequent smaller amoimt of wear in
the crank bearings and joints. And, further, due to a more uni-
form thrust, throughout the cycle, there is a smaller fluctuation
in speed; hence, for the same imiformity of speed, the fly-wheel for
a compoimd engine need not be as massive as that for a single-
expansion engine. On the other hand, the compound engine has
a greater number of moving parts; hence, a greater first cost,
and additional friction.
COMPOUND ENGINES 183
It is obvious that the volume of the low-pressure cylinder of a
compound engine, for the same initial and final volumes, must
be the same as that of a single-expansion engine. The volume
of the low-pressure cylinder is therefore fixed by the boiler pressure
of the steam, the total expansion, the power to be developed, and
the speed of the engine. The volume of the high-pressure cylinder,
on the other hand, is a matter of choice; provided always that the
ratio of the volume of the low-pressure cylinder to that of the
high-pressure cylinder is less than the total ratio of expansion.
The point of cut-oflF, however, for the high-pressure cylinder,
as will be shown later, depends upon the ratio of the two volumes.
From a thermodynamic standpoint, the ranges in temperature
for the two cylinders should be about equal; since this gives equal
fluctuations of temperature in the cylinders. This also gives,
very nearly, equal amounts of work done in the two cylinders;
which, as will be seen later, is also best mechanically. Since it is
advisable to have nearly equal ranges of temperature in the two
cylinders, it necessarily follows that the ratio of the cylinder
volumes is fixed by the total ratio of expansion. In practice,
depending upon the total ratio of expansion, the ratio of the
volume of the low-pressure cylinder to that of the high-pressure
cylinder may vary from 3 to 5.
141. Tandem Conq^ound Engine with Large Receiver. By a
tandem compound engine, is meant an engine which has the axes
of the two cylinders aligned; and has only one piston rod, which
carries both pistons. In an engine of this type,. the two pistons,
necessarily, have strokes of equal lengths. Assume the volume
of the receiver to be so large in comparison with the volume of the
two cylinders, that there are, during the cycle, no fluctuations of
pressure in the receiver. If, further, there be assumed no clear-
ance and no losses whatsoever, then the indicator diagram will
be identical with that depicted in Fig. 33.
Let, in Fig. 34, H represent the high-pressure and L the low-
pressiu'e cylinder; and, further, let R be the ratio of the volume
184
THERMODYNAMICS
of L to that of H, If pi be the pressure, during admission, of the
steam in Hj and p2 the pressure of the steam in the receiver,
which is also the back pressure on Pi, the piston of Hy then, neglect-
ing the area of the rod, the stress in the rod, between the two
cylinders, due to these two pressures, is
5i = (pi-p2Mi;
(1)
where A\ is the area of Pi, and Si the stress. In a similar manner,
the stress in the rod to the right of P2, due to the pressures p2
and p3, in the cylinder L, is
iS2 = (p2-p3)A2;
(2)
P« L
Pi H
t
^pr
,
*-- p, — -
'K
•*- P3- -*
Fig. 34.
where A2 is the area of P2, and 82 the stress. Taking the sum
of the right-hand members of equations (1) and (2), we obtain
for the stress, in the rod r,
S=(pi-P2M1 + (P2-P3M2 (3)
Since now, the cylinders are of equal length, and R is the ratio
of their volumes, we have
Ai^A2/R.
(4)
Substituting the value of A 1, as given by equation (4), in equation
(3), we obtain
A2
S=(P1-P2)^ + (P2-P3)A2.
. . (5)
COMPOUND ENGINES 185
Had the expansion taken place in the low-pressure cylinder
between the pressures pi and pa, then the stress in the rod would
have been found to be
S' = (pi-P3M2. . . . ^. . . (6)
Since the relation of pressure and volume of steam is not
expressible by a simple equation, it is impossible to eliminate
P2 from equation (5); and, therefore, no general comparison
between S, as given by equation (5), and S', as given by equation
(6), can be made. But, by assuming particular values for pi
and p2f some idea may be obtained in regard to the relation of
the stresses S and S\
To make a comparison, assmne pi to be 100 lbs., and pa 1
lb. ; and, as a matter of convenience, p2 to be 20 lbs., then p2 = Pi/5.
Substituting this value of p2 in equation (5), we find
Again, for the case under consideration, R will have a value
of about 3; hence, equation (7) becomes
S^fxf+^-psA,=^(^-^-p^)A,. . . (8)
By comparing equations (6) and (8), it is seen that for the
same given initial and final pressures, the maximmn thrust for the
single-expansion engine is more than double the maximum thrust
for the double-expansion engine.
The indicator diagram, of Fig. 33, represents an extreme case;
and one that cannot be realized in practice. For, in the first
place, to maintain a constant pressure, during the exhaust of the
small cylinder and the admission to the large cylinder, requires
a receiver of excessive bulk. Secondly, there is always a certain
amount of resistance oflfered to the flow of steam in passing from
the first cylinder to the receiver, and from the receiver to the second
186 THERMODYNAMICS
cylinder. Therefore, the lines representing, respectively, the
exhaust for the small cylinder and the admission for the large
cylinder, will not coincide. That is, the exhaust line, for the
small cylinder, will show a higher pressure than that shown by the
admission line for the large cylinder. This drop in pressure is,
however, not entirely wasteful; since, due to this partially unre-
sisted adiabatic expansion, part, and in some cases all, of the
moisture formed in the first cylinder is removed.
142. Compound Expansion without Receiver* In some
engines, usually called Woolf engines, the steam passes directly
from the one cylinder to the other. The cylinders may be either
in tandem or side by side. It is obvious that in engines of this
type the two pitsons must begin and end their strokes together.
That is, the movements of the two pistons must either be in phase
or differ by 180°. The operation is then as follows: Steam is
admitted to the small cylinder, up to some desired fractional part
of the stroke when cut-off takes place, and then expands to the
end of the stroke. At the end of the stroke communication is
established between the two cylinders, the steam begins to pass
from the small cylinder to the large cylinder and a second expan-
sion takes place. Since, now, the cylinders must remain in com-
munication to the end of the stroke, there can be no cut-off in
the low-pressure cylinder; and furthermore, since the pressm'es
in the two cylinders are, at any instant, the same, it follows that
the pressure in the small cylinder, during its exhaust, is continually
decreasing. Admission and expansion now again take place in
the small cylinder, while exhaust is taking place in the large
cylinder.
The indicator diagrams, for the cycle just discussed, if it be
assumed that there is no clearance and no drop between the two
cylinders, will be as represented by Fig. 35. For the high-pres-
sure cylinder, AB represents the admission, BC the expansion,
and CD the exhaust; and for the low-pressure cylinder, EF repre-
sents the expansion, FG the drop during release, and GH the
COMPOUND ENGINES
187
exhaust. To combine the two diagrams, the diagram EFGH must
be drawn for a piston area corresponding to that of the diagram
ABCDy so that equal increments in abscissas, for the two diagrams.
A B
B
M — ...^^^
I ]^:sfL.
K L,/.
"^"""---^F
u
^
n
Vr
V
Fig. 35.
measure equal increments in volume. That part of the indicator
diagram of the high-pressure cylinder, represented by ABCI,
will then remain unchanged for the combined diagram. And,
to find any point on the expansion curve, for the combined diagram
A B
I J\a
V
Fig. 36.
beyond the point C, corresponding to the pressure as represented
by the point K^ it is only necessary to draw a horizontal line,
such as KN, and remember that the total volmne of the steam,
for this pressure is equal to KL+MN. This is represented in
Fig. 36, by the line KN, In a similar manner, a number of points
188
THERMODYNAMICS
are found, and the expansion curve BCF is determined. This
gives, then, for the equivalent indicator diagram, for the two
cylinders, the diagram ABCFGH.
143. Tandem Compound and Small Receiver* Since it is
impracticable to have a receiver of sufficient volume, so that the
pressure in it is constant throughout the cycle, the cut-oflF, for the
large cylinder, must be so chosen that the pressiu-e in the receiver,
at the end of the exhaust stroke of the small cylinder is the same
as when release occurs, during the next stroke in this cylinder.
Let, in Fig. 37, ABCDE be the combined diagram, constructed
as described in Art 142; AB represents the admission to the small
cylinder, BC the total expansion, CD the drop, during release,
in the large cylinder, and DE the final exhaust.
Let, now, F be the point of release for the small cylinder, then
at the same instant admission takes place in the large cylinder;
and, since the rate of volumetric displacement for the piston of the
large cylinder is greater than that for the piston of the small
cylinder, the pressure in the receiver must fall. Let FG represent
that part of the exhaust curve, for the small cylinder, before cut-oflF
takes place in the large cylinder. After cut-oflf has taken place
in the large cylinder, the steam remaining in the small cylinder is
compressed, as represented by the curve GHj into the receiver.
h
COMPOUND ENGINES 189
If, now, there is to be no drop in pressure when release occurs in
the small cylinder, the pressures for the points F and H must be
equal. Since H represents the pressure of the steam, when admis-
sion begins in the large cylinder, then, by beginning with this
point, and taking into account the total volume of the steam for
various points, a curve such as HI, which shows the falling oflF
in pressure in the large cylinder during admission, is found. And,
where this curve, HI, cuts the curve of expansion BC, will be the
point of cut-off for the large cylinder; the pressure at / and G
being identical.
As a matter of cofivenience, the foregoing discussion has been
made with the assumption that the cylinders have no clearance.
No new difficulty, however, is introduced by considering the clear-
ance. As previously explained, a small amount of drop is not
harmful; hence, absolute precision is not required.
144. Cross-compotind Engines. The type of double-expan-
sion engines most frequently used, are those known as cross-corn-
pound engines. Cross-compound engines are either in twin, i.e.,
the cylinders are side by side, and the cranks make an angle of
90° with each other, or else, the cylinders make an angle of 90°
with each other, and the connecting rods act upon cranks in the
same phase. In either case the piston movements are not in
phase. That is, there is a phase difference of 90°; therefore,
when exhaust begins to take place from the small cylinder, the
piston of the large cylinder will not be in a position such that
steam can be received. Hence a receiver is necessary.
From a mechanical standpoint, the cross-compound engine
is far superior to the tandem-compoimd; for, in the one case the
cranks are actually at right angles, and in the other case, where
the cylinders are at right angles, the mechanical effect is precisely
the same as though the cranks made an angle of 90° with each
other. Therefore, the turning moment, throughout the cycle, is
much more uniformly distributed.
The method employed to determine the point of cut-off for
190 THERMODYNAMICS
the large cylinder, in a cross-<;ompoiLnd engine, so that there shall
be no drop, is almost precisely the same as that discussed in Art.
143. It is only necessary to take into consideration the phase
relation of the two piston movements.
In large steam-power plants, the cross-compound engine is
the one most commonly employed; boiler pressures as high as
200 lbs. to 250 lbs. being, in some cases, used. Within the last
few years, compound engines have been operated in conjunction
with low-pressure steam turbines; the turbine operating on the
exhaust steam from the low-pressure cylinder.
146. Triple Expansion. Where a more uniform turning
moment, than that offered by a cross-compound engine, is desired,
triple expansion is employed. In the case of triple-expansion
engines, the cranks are frequently set so that each crank differs
in phase by 120** from the other two; thus giving a good distribu-
tion of turning moment. In other cases, however, the triple
expansion takes place in four cylinders; i.e., one high-pressure
cylinder, one intermediate cylinder, and two low-pressure cylin-
ders. Both of these low-pressure cylinders take steam from the
receiver, to which the intermediate cylinder rejects. The engine is
equipped with four cranks, with a continuous phase difference
of 90°. Triple-expansion engines have been very largely used
in marine engineering; but, are now being superseded either by
turbines, or else by the combination of triple-expansion engines
and low-pressure turbines, which operate on the low-pressure
exhaust steam from the -reciprocating engines.
From the foregoing, it is obvious that, for cross-compound
and triple-expansion engines, it is desirable to have the work
done in the various cylinders equal; since this gives the most
nearly uniform turning moment for the entire cycle. And,
as has been previously stated, the ranges of temperature, for the
various cylinders, will be nearly equal, when the work done in the
cylinders is equal.
Engines having more than three stages are rare; and, it is
COMPOUND ENGINES 191
doubtful whether they are ever economical. Theory can merely
serve as a guide; the final criterion being experiment. In any
given case, however, there is a limiting value for the nimiber of
stages of expansion; and, in practice, this is fixed when the ther-
modynamic gain is offset by the interest and depreciation of
the extra capital invested, plus the extra mechanical losses
146. Tests of Performance. The tests on compound engines
are very similar to those on simple engines. However, when
making tests on an engine, for efficiencies, the load should be varied
from zero load up to, say, 25 to 60 per cent overload, and a curve
plotted, efficiencies as ordinates and loads as abscissas. Or,
if it be desired, the number of pounds of water per B.H.P., or
else the number of B.T.U. per B.H.P., may be plotted against
B.H.P. The curves may also be plotted, using I.H.P. instead
of B.H.P. In any case, however, the curve will show the character
of performance of the engine for the various loads; and, for what
load the best economy is obtained.
If the curve shows that the efficiency of the engine does not
decreaije rapidly, as the load is decreased, from the normal, then
good service will be obtained for variable loads. On the other
hand, if the efficiency falls off rapidly with decreasing load, then
the engine will give good service only for approximately constant
loads.
CHAPTER XIV
INTERNAL COMBUSTION ENGINES AND FUELS
147. Whereas, in the steam engine, the combustion of the
fuel and the application of heat to the working substance, take
place outside of the engine, and in the internal combustion engine,
as the name implies, combustion of the fuel, and the application
of heat to the working substance, take place directly inside of the
cylinder, there is, between the two types of heat motors, as regards
the manner in which the application of heat takes place, a fim-
damental difference. Some of the other prominent differences
of operation between the two types of heat engines will be discussed
later.
The most common fuels which may be used in internal com-
bustion engines are: Coal-oils, alcohol, natural gas, producer gas,
blast-furnace and coke-oven gas, city illuminating gas, etc. The
fuels most generally used are: Producer gas, natural gas, gasoline,
petroleum and alcohol; and in all cases, there must be present
a proper amount of air, so that suflBcient oxygen is supplied, to
bring about complete combustion. Which of these fuels is best
depends upon a great many factors; principally upon duty,
economy, and convenience.
There are three typical methods for the operation of internal
combustion engines; it being the aim to bring about the applica-
tion of heat to the working substance, for the three different
types, respectively, at constant volume, at constant pressure, and
at constant temperature; thus, giving three distinct cycles, which
will now be discussed in detail.
192
INTERNAL COMBUSTION ENGINES AND FUELS 193
148. Four-phase Cycle. The four-phase * or " Otto " cycle
was first applied to the internal combustion engine by Dr. Otto,
in 1876; and is essentially as follows: (1) The piston is at the
end of a return stroke with the exhav^trvalve just closed and the
inlet-valve open to a chamber where the mixing of the fuel takes
place. The mixture consists either of gas and air, or, if a liquid
fuel be used, of vaporized liquid and air. The piston then makes
an outward stroke, called the aspirating stroke, and a charge of
mixture is forced, by the external pressmre, into the cylinder.
(2) The inlet-valve is closed, a return stroke takes place, called
the compression stroke, and the mixture is compressed until its
volume is reduced to that of the clearance. (3) The charge is now
ignited, usually by an electric spark, and combustion takes place
very rapidly, and practically at constant volume, together with
a rapid augmentation of pressure. The gaseous mixture then
expands, forces the piston outward and does work on it. This
stroke is called the working stroke or power stroke. (4) When
the piston is at the end of the working stroke the exhaust-valve
opens; and, during the return stroke, called the expulsion stroke,
the products of combustion are expelled, and the cycle is com-
pleted.
It will be remembered that, in the reciprocating steam engine,
every other stroke is a working stroke. Hence, the internal
combustion engine, operating on a four-phase cycle, having one
working stroke only- for every four strokes, must necessarily
during the working stroke, other things being equal, store more
energy in the fly-wheel. The fly-wheel must, therefore, have a
greater moment of inertia in order to carry the load properly
during the remainder of the cycle.
The ideal indicator diagram, for a four-phase cycle is represented
in Fig. 38. OF is the axis of zero pressure, and AB represents
the aspirating stroke; the pressure in the cylinder and that of
♦Ordinarily called "four-cycle"; this, however, is not proper, since
there are four phases to a cycle.
194
THERMODYNAMICS
the atmosphere being identical. BC represents the adiabatic
compression of the charge, and the vertical line, CD, the com-
bustion at constant volume; OG being the axis of zero volume.
DE represents the adiabatic expansion of the gaseous mixture
after combustion, EB the drop in pressure to that of the atmos-
phere, when the exhaust-valve opens, and BA represents the expul-
sion of the products of combustion. The net work done, by the
working substance, dining the cycle, is represented by the area
DEBC,
O V F
Fio. 38.
149. Two-phase Cycle. In the two-phase cyde* as in the four-
phase cycle, it is the aim to bring about the application of heat,
to the working substance, at constant volume. In the two-phase
cycle engine, however, every other stroke is a working stroke.
Beginning with the ignition of the mixture, there is a rapid
rise in pressure, and then expansion of the gaseous products
of combustion. In the meantime, a new mixture has been slightly
compressed, either by means of an auxiliary compressor, and forced
into a subsidiary reservoir, or else compressed in the crank case
of the engine. Just before the piston reaches the end of the
working stroke, an exhaust-port is opened and then, immediately
following this, the inlet-port is opened. Since, at the opening
♦Ordinarily called "two-cycle"; which, however, is an improper
designation.
INTERNAL COMBUSTION ENGINES AND FUELS 195
of the exhaust-port, the pressure in the cylinder is slightly in excess
of the atmospheric pressure, the gaseous products of combustion
inmiediately begin to flow, out of the cylinder, through the exhaust-
port. Immediately after the exhaust-port opens, the inlet-port,
on the opposite side of the cylinder, is uncovered by the piston;
and, since the mixtiu-ehas been precompressed to a pressureslightly
higher than that obtaining in the cylinder, at the end of the working
stroke, a new charge flows into the cylinder. By a suitable arrange-
ment the inflowing gas is directed so as to help expel the exhaust
gases. The return stroke now begins; the piston closing first the
inlet-port and immediately following the exhaust-port, and the
mixture is compressed until its volume is reduced to that of the
clearance, when ignition takes place, and the cycle is completed.
From the standpoint of thermal efficiency, the four-phase
cycle engine is superior to the two-phase cycle engine; but, on
the other hand, the latter engine is far simpler in construction,
especially in valve gearing, than the former. And, for the same
power output, the two-phase cycle engine always has a smaller
mass than the four-phase cycle engine. Hence, for the same power,
the two-phase cycle engine requires less space than does the four-
phase cycle engine. The thermal efficiency of the two-phase
cycle engine is affected by the fact that the products of combus-
tion are more or less imperfectly expelled; and sometimes, due
to partial mixing in the cylinder of the new charge and the products
of combustion, the exhaust gases may contain unbumt fuel.
Furthermore, it may happen that complete combustion has not
taken place when the inlet-port is opened, thus causing premature
ignition.
160. The Brayton Cycle. In the BrayUm cycle, it is aimed to
bring about the application of heat at constant pressure. In
its operation the Brayton engine compresses air in a separate
cylinder and stores it in a receiver. This compressed air is admit-
ted through a mass of felt, charged with crude petroleum, to the
cylinder of the engine. An auxiliary valve is continuously open,
196 THERMODYNAMICS
permitting a very small jet of air, charged with petroleum, to
flow into a small chamber communicating with the cylinder,
where it bums continuously while the engine is in operation.
When the main inlet-valve opens, air rushes through the felt,
takes up a charge of petroleum, and is ignited by the small flame.
Since, now, there is direct communication between the cylinder
and the air reservoir, the pressure in the cylinder cannot rise above
that in the reservoir; and therefore, combustion takes place at
practically constant pressure. The fuel supply is then cut oflF,
and expansion takes place, approximately adiabatically, until
the pressure has fallen, depending upon valve adjustment, to
any desired value. The exhaust-valve is then opened, and
during the return stroke the products of combustion are expelled;
hence, this cycle is a two-phase cycle. The Brayton engine has
practically been superseded by the one which will be discussed
in the next article.
161. The Diesel Cycle. In the Diesel engine the aim is to
bring aboiit combustion, and therefore the application of heat,
at constant temperature. The Diesel cycle is as follows: During
the aspirating stroke a charge of air flows into the cylinder, where
it is compressed, approximately adiabatically, to a very high
pressure, during the compression stroke. The inlet-valve now
opens and the fuel in the form of oil, usually crude oil, is injected
and immediately becomes ignited, due to the high temperature
of the air, caused by the high precompression. Combustion now
takes place at nearly constant temperature, the piston advances,
and the expansion of the gases, up to the point when the fuel
is cut off, is nearly isothermal. From this point, up to where the
exhaust-valve opens, the expansion is approximately adiabatic.
The exhaust-valve being fully open, the pressure falls to that of
the atmosphere, the expulsion stroke takes place, and the cycle
is completed. The Diesel cycle, like the Otto cycle, is a four-
phase cycle.
The ideal indicator diagram, of the Diesel engine is shown in
INTERNAL COMBUSTION ENGINES AND FUELS 197
Fig. 39, in which AB represents the aspirating stroke, BC the
adiabatic compression, CD the isothermal expansion, DE the
adiabatic expansion, EB the drop in pressure when the exhaust-
valve opens, and BA the expulsion stroke. Theoretically, the
adiabatic expansion DE may be carried up to a point when the
pressure of the gases has fallen to that of the atmosphere. But,
for the same reason as was given in discussing the steam engine,
in Art. 115, this is not economical.
Theoretically, the Diesel cycle is the most efficient cycle, so far
used, in the operation of internal combustion engines; realizing,
V
FiQ. 39.
as it does, the application of heat at a practically constant temper-
ature. Thus approaching, as regards the absorption of heat, the
Camot cycle. The high precompression, however, requires that
the engine have a very massive fly-wheel, and subjects it to severe
strains. Furthermore, to start the engine, a reservoir of com-
pressed air is required, operating on it very much as steam operates
on a steam engine; and when the engine is up to speed, the valves
are shifted, so as to disconnect the air, and £ulmit the oil. How-
ever, by means of Diesel motors of 300 H.P. capacity, thermal
efficiencies as high as 32 to 33 per cent have been obtained.
198 THERMODYNAMICS
Fuels and Fuel Tests
162. Before discussing, mathematicallyy the ideal indicator dia-
gram of the internal combustion engine, it will prove instructive
to consider briefly the chemical behavior of the fuel as combus- ^
tion takes place; and also to compare the volume of the mixture,
before combustion, with the volume of the products of com-
bustion. For the volumetric comparison, the same temperature
and pressure must, of course, be assumed before and after combus-
tion; and we must know what volume the fuel, in the vaporized or
gaseous condition, will occupy. Furthermore, from the chemical
constitution of the fuel, we must determine the quantity of oxygen
which has to be supplied to bring about complete combustion,
and the volmnetric changes due to changes in chemical con-
stitution.
163. Chemical Reactions. Experiment shows that, when two
or more gases react chemically to form a gas or gases of different
chemical constitution, the numbers, representing the volumes of
the combining gases, are fixed with respect to each other by definite
simple ratios; and, likewise, the volume or volumes obtained,
after the chemical reaction has taken place, are definitely fixed by
the volmnes of the combining gases. The foregoing is best illus-
trated by the consideration of a few concrete cases. As an
example, when the gases hydrogen and oxygen react chemically
to form steam, then for every given volume of oxygen there is
required double the volume of hydrogen; or, to put it numer-
ically, two liters of hydrogen combine with one liter of oxygen
to form two liters of dry steam. In the notation adopted by
chemists, this chemical action is expressed as follows:
H2+0 = H20 (1)
The volmnetric relations are, of course, only true for identical
temperatures and pressures. Likewise, one liter of carbon vapor
INTERNAL COMBUSTION ENGINES AND FUELS 199
unites with an equal volume of oxygen to form two liters of
carbon-monoxide; and the two liters of carbon-monoxide will
combine with one liter of oxygen to form two liters of carbon-
dioxide. The reaction is expressed symbolically by
C0-f-0 = C02 (2)
Finally, as another example, six volumes of carbon vapor unite
with six volmnes of hydrogen to form two volmnes of benzene;
which is given by
C6+H6 = C6H6 (3)
It will be noticed that in all cases, excepting that of two ele-
mentary gases of equal volmnes, there is, when the gases unite
chemically, a reduction in volume; but, in every case, and this
is generally so, the volume of the combined gases is two units
of the measure of volume chosen. It must, however, not be under-
stood, from the foregoing equations, that in all cases the compound
is formed in as simple a manner as here expressed; for, interme-
diate steps are frequently necessary.
It is of interest to note that, if the density of one of the gases
be chosen as unity, and the relative densities of the various gases
be known, the density of the compound gas may immediately
be found from its chemical formula.* Thus, if the density of
hydrogen be taken as unity, which is the most convenient, its
density being less than that of any other gas, then the density
of oxygen, for the same temperature and pressure, is approximately
16. Hence, we find, from equation (1), since three volumes,
having a combined mass of 18, reduce to two volumes, the density
of dry steam, with respect to hydrogen equals 9. In a similar
manner, from equation (2), since carbon vapor has 12 times the
* The determination of the chemical constitution of compounds is
usually attended by extreme difficulties; but it is here neither possible nor
is it essential to describe the various methods used. For complete descrip-
tions and discussions, the student is referred to standard works on physical
chemistry.
200 THERMODYNAMICS
density of hydrogen, the density of carbon-dioxide is found to
be 22; and, from equation (3), the density of benzene is found to
be 39.
IM. Gasoline. The most volatile of all the fuel oils obtained
from petroleimi, by fractional distillation, is gasoline. The com-
position of gasoline is somewhat variable; but, its chemical con-
stitution is represented with sufficient accuracy by the formula
CeHH, and its density by 0.7. Assimaing the formula C6H14,
then the chemical equation, representing complete combustion,
is given by
C6Hi4+190=6C02+7H20 (4)
Since the temperature in the cylinder of an internal com-
bustion engine is of sufficient intensity to insure complete vapor-
ization of the fuel, the gasoline vapor will behave as a gas; and
since two volumes of gasoline vapor will yield, upon decomposi-
tion, six volimaes of carbon and fourteen volumes of hydrogen,
nineteen volumes of oxygen, under the same condition, as regards
temperature and pressure, must be supplied to bring about com-
plete combustion. Twelve of these volumes of oxygen will com-
bine with the carbon, to form twelve volumes of carbon-dioxide,
and seven volumes will combine with the hydrogen to form four-
teen volumes of dry steam. Since the average composition of
the air, by volume, is 21 per cent oxygen and 79 per cent nitrogen,
it follows that for every volume of oxygen supplied there will be
present 79/21 volumes of nitrogen. Hence, assuming that the
mixture, before combustion, contains two volumes of gasoline
vapor, then its total volume is
79
2+19+xtX19 = 92.5 volumes (very nearly).
And after combustion the volimae, referred to the same temper-
ature and pressure, is
79
12+14+— X 19 = 97.5 volumes (very nearly).
INTERNAL COMBUSTION ENGINES AND FUELS 201
Hence, the volume of the products of combustion, referred to the
same temperature and pressure, if just enough air be present to
bring about complete combustion, is approximately 5.4 per cent
in excess of the volume of the mixture. Experience, however,
shows that, for an internal combustion engine to operate satis-
factorily, the quantity of air supplied to it must be considerably
in excess of that required for complete combustion. The excess
may vary from 15 to 50 per cent; and this, together with the neu-
tral gases, remaining in the cylinder after the expulsion stroke,
reduces the difference between the volumes, before and after
combustion, appreciably.
166. Kerosene. The most important of the fuel oils, obtained
by the fractional distillation of petroleum, and representing about
50 per cent of the total yield, is kerosene. Kerosene is consider-
ably less volatile than is gasoline, has an average density of about
0.805; and its chemical constitution is represented, with a fair
degree of accuracy, by the formula C10H22. The chemical equa-
tion, representing complete combustion, then is
CioH22+310 = 10C02+llH20; . ... (5)
and by the method used, when dealing with gasoline, we find for
the volume of the mixture, before combustion,
79
2+31+^X31 = 149.6 volumes (very nearly).
After combustion, the volume, referred to the same temperature
and pressure, is
79
20+22+2tX31 = 158.6 volumes (very nearly).
Hence, if the quantity of air present carries just a sufficient amount
of oxygen to bring about complete combustion, then the volume
of the products of combustion is 6 per cent in excess of the volume
202 THERMODYNAMICS
of the mixture. This percentage difference is, of course, reduced
somewhat by an excess of air and neutral gases being present
during the cycle.
That the volumes of the gases, before and after combustion,
are found to be nearly equal is due to the fact that the volume
of inert nitrogen, which is necessarily present, is always large in
comparison with the volume of the other gases. In the two cases
just discussed, viz, gasoline and kerosene, the volume of the gases
after combustion was found to be greater than the volume of the
mixture. This, however, is not necessarily the case; for, if
we consider carbon-monoxide as a fuel we shall find conditions
reversed. The chemical equation, expressing complete com-
bustion for CO, is
C0+0 = C02; (6)
from which we find, respectively, for the volumes of the mixture and
products of combustion, approximately 6.8 and 5.8. This gives
the volume, after combustion, approximately 14.7 per cent less
than the volume of the mixture. CO is a gas of much lower
density than is either gasoline or kerosene vapor; and, by consider-
ing the two following cases, it becomes manifest that, other things
being equal, the higher the density of the fuel gas, the greater the
volume of the gases, after combustion, in comparison with the
volume of the mixture. Assume the two fuels to be, respectively,
C2H2 and CeHe. For the former we have
7Q TQ
C2H2+50+gx5N = 2C02+H20-hgx5N; . . (7)
and, for the latter,
7Q * 7Q
C6H6+150+gxi5N = 6C02+3H20+^Xl5N. . (8)
From equation (7), we find, for the volumes, before and after com-
bustion, respectively, 25.8 and 24.8; and, from equation (8) we
find, for the volmnes, before and after combustion, respectively,
INTERNAL COMBUSTION ENGINES AND FUELS 203
73.4 and 74.4. That is, in the latter case, where the density of
the fuel is greater, the volume of the gases after combustion, in
comparison with the volume of the mixture, is greater than it
is in the former, for the less dense fuel. In general, the fuel con-
sists principally of carbon and hydrogen; and the products of
combustion consist of CO2 and H2O vapors. Hence, if the fuel
is comparatively rich in hydrogen, then, since the H2O vapor
is of a much lower density than is CO2 vapor, it follows that the
products of combustion occupy a greater volume than they would,
were they produced, from a fuel poor in hydrogen.
166. City Gas. The gases, used for illuminating purposes,
in different cities, vary considerably in composition. Not only
is there a variation in going from one plant to another, but the gas
drawn from the supply main will be found to vary somewhat for
different parts of the day. The following table is a fair average
for the composition, by volume, of the gas supplied in the Borough
of Manhattan:
Per Cent.
Carbon-dioxide (CO2) 1.9
Illuminants (practically C2H4) 9.9
Carbon-monoxide (CO) 18.2
Methane (CH4) 22.75
Hydrogen (H2) 42.
Nitrogen (N2) 5.25
It is, therefore, impossible to make computations, of any value,
with respect to city gas, imless samples of the gas supplied to the
engine, are subjected to chemical analysis.
157. Calorific Value of Fuel. In general, to decompose a
compound, into its constituents, requires an expenditure of energy
in the form of heat; hence, unless the fuel is in a form such that
it can combine directly with oxj^gen, without first being decom-
posed, the available heat, i.e., the calorific value of the fuel, is less
than that developed by its combination with oxygen. Further-
204 THERMODYNAMICS
more, if the fuel is in the liquid form, heat is absorbed to convert
it into a vapor. In general, however, the heat of vaporization is
negligibly small in comparison with the other quantities involved.
As a matter of illustration, we will assume that methane (CH4)
is used as a fuel. The chemical equation, for complete com-
bustion, is
CH4+40 = C02+2H20 (9)
In Art. 29, it was stated that when 1 gram of H combines with
to form H20,- about 34,000 gram calories are evolved, and dur-
ing the combination of 1 gram of C with to form CO2, about
8000 gram calories are evolved. Assuming now, as a matter of
convenience, 1 gram of CH4, then since the density of carbon
vapor with respect to hydrogen is 12, we will have 0.25 grams of
H and 0.75 grams of C. Therefore, the complete combustion
of 1 gram of CH4, if there were no heat required to decompose
the compound, would yield
0.25X34,000+0.75X8000 = 14,500 gram calories.
Experiment, however, shows that, when 1 gram of CH4 is con-
smned, to form H2O and CO2, approximately 13,200 gram calories
are evolved. Hence, the diflference, viz, 1300 gram calories are
absorbed in decomposing the compound into its elements.
It must always be remembered that, the experiments conducted
for the purpose of determining heats of combustion and decom-
position of a fuel, though simple in operation, may be attended
by difficulties of a chemical nature; and therefore, in general,
the values found in tables, must be taken as being only approx-
imations.
168. Determination of Calorific Values. The determination
of the calorific value of a gaseous fuel is a very simple experiment,
provided there is available a modem gas calorimeter. The gas,
from the source of supply, is first passed through an accurately
INTERNAL COMBUSTION ENGINES AND FUELS 205
calibrated meter, and from this through a pressure regulator which
maintains, throughout the progress of the experiment, the pressure
of the gas constant. From the pressure regulator, the gas is
supplied to a Bunsen burner where combustion takes i^lace.
The air has free access to the chamber in which the burner is
placed, such that a suflBcient quantity of oxygen is supplied to
insiu*e complete combustion. From the combustion chamber, the
products of combustion pass through a system of tubing, sur-
roimded by circulating water, in a manner such that the tempera-
ture of the exhaust gases is reduced to that of the outflowing
water, which is practically the same as that of the surroundings,
when exhaust takes place. The inflowing water, by means of
proper arrangements, is maintained at practically a constant pres-
sure and temperature; and hence, when steady conditions have
been attained, the temperature of the outflowing water will also
be practically constant.
The exhaust gases will consist mainly of CO2; since the H2O,
which is formed by combustion, is condensed. After steady
conditions have been assmned, i.e., the water due to condensa-
tion is flowing at a constant rate, and the temperatures of the water
at inflow and outflow are sensibly constant, the outflowing cir-
culating water is caused to flow into a receptacle of known weight,
and at the same time the water of condensation is caused to flow
into a second receptacle of known weight. Simultaneously with
these two operations, a reading is taken on the gas meter and the
temperatures of the circulating water at inflow and outflow are
noted. The temperatures of the water at inflow and outflow
are then noted at suitable intervals of time, imtil a suflScient
quantity of circulating water has been collected. A second
reading is then taken on the gas meter, and, simultaneously with
this, the collection of water in the two receptacles is discontinued.
From the initial and final readings of the gas meter, the volume
of the gas consumed, corresponding to the existing pressure and
temperature, is found; and this volume, corrected to a tempera-
206 THERMODYNAMICS
tiire of 62°F.* and a pressure equal to 30 inches of mercury,
gives the volume, as regards American gas engine practice, for
standard conditions. From the mean of the temperatures for
inflow and outflow, together with the quantity of circulating
water collected, the quantity of heat, absorbed by the circulating
water, is readily found.
Let during the progress of the experiment, ti and T2, respect-
ively, be the mean of the temperatures of inflow and outflow, and
M the mass of circulating water collected, then the quantity of
heat absorbed, by the circulating water, is
Q' = ilf(T2-Tl) (10)
Q', in equation (10), represents the heat of combustion, plus the
heat of condensation. Since, however, the temperature of the
exhaust gases, in the case of any internal combustion engine,
is always higher than that of condensation, corresponding to
the existing pressure, of the H2O vapor present, it follows that
the heat available for doing work in the engine is less, for the
given quantity of gas consumed, than is indicated by equation
(10). To obtain the available heat, there must be deducted from
the heat absorbed by the circulating water, the heat due to con-
densation, plus the heat given up by the condensed steam in
cooling from the temperature of condensation to the temperature
of the outflowing water. Let m be the mass of steam condensed,
T the temperature at which condensation takes place, and r the
corresponding heat vaporization. The quantity of heat given
up to the circulating water, by the steam and condensed water,
then is
g=7n!r+(T:— T2)! (11)
* There is no gain by using 62 °F. in place of 0°C. For, the temperature
is seldom 62 °F.; hence, if accurate results are required, a correction for
temperature is necessary. It therefore would be more convenient if 0°C.
were at all times used, since this is the temperature used by physicists and
chemists, as the standard temperature, for which the volumes of gases are
specified.
INTERNAL COMBUSTION ENGINES AND FUELS 207
Subtracting equation (11) from equation (10), we find the avail-
able heat,
Q=Jlf(T2— ci)-m{r+(T-T2)} (12)
Dividing by F, the volume of gas consumed, we find the available
heat per irnit volimie; i.e., the calorific value, is
^^ Jf(T2-Tl)~mir+(T-T2){ ^ ^ ^ ^ /J3X
The calorific value is usually specified in B.T.U. per standard
cubic foot of gas.
The temperature of condensation is approximately the same
as the normal boiling-point, since, the pressure in the calorimeter
does not vary greatly from that of the atmosphere. However,
since q is always small in comparison with Q\ no serious error
can be introduced in the final result by not estimating the value
of T with absolute precision.
169. Liquid Fuels. In the case of liquid fuels, the calorific
value may be determined in precisely the same manner as that of
gases. It is only necessary to have an accessory piece of appar-
atus, by means of which the liquid is converted into a gas or vapor;
the quantity of fuel consumed being determined by weighing.
In the case of liquid fuels, however, the calorific value is usually
specified in B.T.Us. per pound of fuel.
As a matter of interest, a table is here appended, in which
is given the lower calorific values of various substances; i.e., that
calorific value is given for each substance which would have to
be taken if it were considered as a fuel for an internal combustion
engine. The calorific values, given in this table, are approx-
imately correct for the first three figures; and the chemical
formulae for gasoline and kerosene, as previously stated, are only
approximations. For gaseous mixtures, such as city gas, no
figures for the calorific value can be given on account of the vari-
208
THERMODYNAMICS
ablity in constitution. Actual experiment, by means of a Junker's
calorimeter, shows the calorific value of illuminating gas as fur-
nished to the Borough of Manhattan, to be about 590 B.T.U.
per cubic foot.
CALORIFIC VALUES
IBritish Thermal Units.
Gas.
Chemical
Formula.
Gram
Calories,
Per Gram.
Per Pound.
Per Cu. Ft.
at 32'* F.
at 62« F.
HvdroKen
(H,)
(CHO
(CU.)
(CH4)
(COI.)
(C4H,)
(C,H.)
(CO)
(CO
(C,^,4)
(CioHa)
(CJH.O)
(CH4O)
28900 .')2n00
291
957
1530
1600
1700
3030
3770
343
273
Methane
11900
11700
11400
11300
10800
9610
2430
8110
10300
10100
6560
4750
21400
21100
20500
20300
19400
17300
4380
14600
18500
18200
11800
8550
898
Acetylene
1440
Ethylene
1510
Ethane
1600
Butylene
2850
Benzene
3540
Carbon-monoxide . .
Carbon
322
Liquid
Gasoline
Kerosene
Ethyl Alcohol
Methyl Alcohol ....
THERMAL CAPACITIES AND DENSITIES OF GASES
Chemical
Formula.
Air
Oxygen
Nitrogen
Hydrogen
Carbon-monoxide . .
Carbon-dioxide ....
Methane
Ethylene
Ammonia
Superheated steam .
(OO
(N,)
(HO
(CO)
(COO
(CHO
(C,H4)
(NH3)
(H2O)
Grams per
c.c. for
Standard
Pressure
at 0'' C.
0.001293
0.001429
0.001255
0.00008955
0.001250
0.001965
0.000715
0.001251
0.0007616
i Thermal Capacities,
Gram Calories Per Gram.
0.2375
0.217
0.244
3.409
0.242
0.217
0.593
0.429
0.530
0.480
C.
0.169
0.154
0.173
2.42
0.173
0.168
0.449
0.346
0.402
0.361
Ratio.
1.405
1.41
1.41
1.41
1.40
1.29
1.32
1.24
1.32
1.33
INTERNAL COMBUSTION ENGINES AND FUELS 209
The values for thermal capacities given in the foregoing table
are the results obtained from experiments conducted, in general,
between the limits of 0**C. and 200°C. In some cases the ranges
were very small; hence, the given values are not necessarily true
for high temperatures.
The various values given in the two foregoing tables have
mostly been taken from Landolt and Bomstein's tables.
CHAPTER XV
IDEAL COEFFICIENT OF CONVERSION AND ELEMENTARY
TESTS
160. In Art. 102 it was shown what fractional part of the heat
abstracted from the source, can be converted into work by a perfect
steam engine and boiler, and in Art 138, a perfect engine was
considered independently of the boiler. Either of these results
may be employed as a standard, depending upon whether we
are considering the engine and boiler jointly or the engine alone.
Similarly, some ideal standard of performance, by means of which
internal combustion engines may be compared, must be assumed.
The ideal indicator diagram based upon the four-phase cycle, has
been found convenient; since this cycle is the one most generally
used.
161. Ideal Indicator Diagram. To deduce an expression
for the maximal amount of work that can be realized during a
cycle, by means of an internal combustion engine, it is, of course,
necessary to assume perfect conditions. The following assump-
tions will be made: The compression of the mixture is adiabatic,
the combustion of the fuel, and therefore the application of heat,
takes place at constant volume, the expansion of the products
of combustion is adiabatic, the rejection of heat takes place at
constant volume, and the thermal capacity of the gas is constant
throughout the process. A further assumption, which by a
previous discussion, under the heading of fuels, has been shown
to be approximately true, will have to be made; viz, that for the
same temperature and pressure, the volume occupied by the
210
IDEAL COEFFICIENT OF CONVEKSION AND TESTS 211
gaseous mixture, before combustion, is equal to the volimie of
the products of combustion.
In Fig. 40, 01 and OH are, respectively, the axes of zero pres-
sure and zero volmne, and the various parts of the cycle are repre-
sented as follows: AS is the aspirating stroke at constant pressure,
BC the adiabatic compression of the mixture, CD the combustion
and application of heat at constant volume, DE the adiabatic
expansion of the products of combustion, EB the rejection of
heat at constant volmne, and BA the expulsion stroke -at con-
stant pressure. Since, during the aspirating and expulsion strokes,
H
D
kT.
C
^ \
\^
^""^H
A
— — ___T4
""■ B
v
FiQ. 40.
the pressures on the two sides of the piston are assumed equal,
the net work done, during the cycle, is obviously measured by the
area DEBC, If, now, we represent by Ti, 7^2, Tz, and Ta, the
temperature of the working substance for the points C, D, E, and
By the heat applied during combustion, while the pressure rises
from that represented by the point C to that represented by the
point Z>, is
Qi=MC.{T2-Ti)]
(1)
where M is the mass of the mixture, Cv the thermal capacity per
unit mass at constant volume, and Qi the heat developed during
combustion. The heat rejected at constant volmne, while the
212 THERMODYNAMICS
pressure falls from that represented by the point E, to that repre-
sented by the point B, is
Q2=MC.{Tz-Ta)\ (2)
where Q2 is the heat rejected by the working substance, while
the temperature changes from Tz to T^. It is immaterial whether
the change in temperature, from Tz to T^, since the final result
is precisely the same, takes place inside or outside of the cylinder.
The cofnditions are analogous to those discussed, in Art. 114, for
condensation at constant volume in the steam cylinder.
Since, now, there is no exchange of heat during the adiabatic
expansion DE, and, likewise, during the adiabatic compression
BCj and the thermal capacity of the mixture is approximately
equal to that of the products of combustion, it follows, from
equations (1) and (2), that the heat converted into work, during
the cycle, is
Qi-Q2^MCUT2-Ti)-{Tz-T^)]. ... (3)
And, since the ratio of the heat converted into work to that
abstracted from the source, is the ideal coefficient of conversion,
we find
_ Qi-Q2 _ MC,{(r2-ri)^(r3-r4)| .
from which
Since both DE and BC represent adiabatic changes for the same
changes in volume, we have, from equation (60), Art. 49,
(V2y^^ ^Tz ^Ta^ Tz-Ta
\vj T2 Ti T2-T1'
where vi is the volume of the mixture before compression, and
V2 the volume after compression. Hence equation (4) becomes
-©■"• («
IDEAL COEFFICIENT OF CONVERSION AND TESTS 213
Equation (5) shows that the ideal coefficient of conversion is
a function of the ratio of the volume before compression to the
volume after compression; and increases with the amount of
precompression.
162. Theoretical Temperatures. The temperature which
would obtain upon complete combustion, if there were no losses,
is readily computed for any given case. That is, the theoretical
rise in temperature, viz, T2—Ti,\3 numerically equal to the ratio
of the heat of combustion to the thermal capacity of the products
of combustion. However, it is found to be necessary, in order
to have proper lubrication between the piston and cylinder
walls, so as to prevent deterioration of material, to abstract heat
from the cylinder walls, either by water jacketingf or else by air
cooling. The former, that is water cooling j is brought about by
having water, at a comparatively low temperatxire, circulate in a
jacket surrounding the cylinder; and the latter, viz, air cooling,
is brought about by increasing the surface of the exposed part of
the cylinder by means of rihsy and having a stream of air plajdng
over it continuously, by means of an air blower of some kind, or
else, as is the case in some automobile engines, the circulation of
air is brought about by the motion of the car. In any case, the
heat abstracted, due to either water or air cooling ^ limits the rise in
temperature. Therefore, the temperature found in the cylinder
of an internal combustion engine, is always less than that pre-
dicted from the heat of combustion and the thermal capacity of
the products of combustion. Frequently, the actual temperature
is found to be only 50 per cent of the theoretical temperature.
163. Standard Diagram. In deducing the expression for the
ideal coefficierU of conversion for the internal combustion engine.
Art, 161, certain assiunptions, in regard to thermal capacities
and volumes before and after combustion, were made, which are
only approximations. But the errors involved in these assmnp-
tions are very small in comparison with the difference between
the actual and theoretical temperatures obtaining in the cylinder.
214 THERMODYNAMICS
However, the diagram described in Art 161, and the results
deduced therefrom, though differing materially from what can
be realized in practice, are very convenient as a basis for com-
paring the performances of internal combustion engines.
Elementary Engine Tests
164. Brake Power and Indicated Power. The power delivered
by an internal combustion engine is determined in precisely the
same manner as is that of a steam engine. This has been fully
described in Art 133. Furthermore, the indicated power of an
internal combustion engine is also found in the same manner as
is that of a steam engine, as described in Art 134. But it must
be emphasized that N, in equation (15) of Art 134, represents
not the number of revolutions per minute of the fly-wheel, but
the number of aydes per minvte in the cylinder under test.
The ratio of Brake Power to Indicated Power is, of course,
in the case of an internal combustion engine, as well as in the case
of a steam engine, a measure of the mechanical efficiency. It is
found, however, that the mechanical efficiency of an internal
combustion engine, other things being equal, is always less than
the mechanical efficiency of a steam engine. This is principally
due to the fact that, owing to the high temperatiu*es existing in
the cylinders of internal combustion engines, the lubrication is
not as good as that obtained in steam cylinders.
166. Thermal Efficiency of Internal Combustion Engine.
The thermal effi^dency of an internal combustion engine is, of course,
the ratio of the power delivered by the engine, to the power due
to the fuel consumed. In making a test, the engine is loaded by
means of a brake, or some other contrivance, to the desired
amount. Then, in the case of a gaseous fuel, the voliune of gas
consmned is measured by means of a meter. Simultaneously with
this, as described in Art. 158, the calorific value of the gas is
determined. The best results are obtained if continuous tests
IDEAL COEFFICIENT OF CONVERSION AND TESTS 215
are made for the calorific value of the fuel; that is, if the
supply to the fuel calorimeter is tapped directly onto the main,
supplying fuel to the engine, and samples of the fuel are tested,
for calorific values, throughout the entire run. The ratio, then,
of the work done by the engine, during the test, to the work,
expressed in the same imits, due to the fuel consumed, which is
equal to the product of the volume of gas consumed during the
run and the mean calorific value of the gas, as found by means of
the gas calorimeter, is a measure of the thermal efficiency. Or,
if a liquid fuel be used, the work due to the fuel consumed, is found
from the product of the mass of liquid consumed, during the run,
and the mean calorific value per imit mass. The calorific value,
per imit mass of the liquid, is determined as described in Art. 159.
166. Actual Indicator Diagram of Intemal Combustion Engine.
By means of the indicator diagram, taken from an intemal com-
bustion engine, the behavior of the working substance may be
conveniently studied. The actual indicator diagram differs,
of course, from the ideal indicator diagram, as depicted in Fig. 40.
Whereas, in the ideal indicator diagram, the line representing
the aspirating stroke, is parallel to the axis of zero pressure, in the
actual indicator diagram the line representing the aspirating
stroke approaches the axis of zero pressure, as represented by
AB in Fig. 41. This is due to the throttling eflfect of the inlet-
valve, on account of which, the pressure in the cylinder decreases
as the piston advances. In Fig. 41, 01 and OH are, respectively,
the axes of zero pressure and of zero volmne, and A A' is the atmos-
pheric line. The curve AB, as just stated, represents the aspira-
ting stroke; and shows the presswe in the cylinder at the end
of this stroke, less than the atmospheric pressure, by an amount
A'B. The compression of the mixture, which is approximately
adiabatic, is represented by the curve BC. The combustion of
the mixture, and consequent rise of pressure in the cylinder, is
represented by the curve CD, which is, if the ignition has been
properly timed, practically parallel to the axis OH. DE is the
216
THERMODYNAMICS
curve representing the expansion of the products of combustion.
At E the exhaust-valve begins to open, and the pressure decreases
rapidly to the end of the stroke F. The expulsion stroke then
begins, and the pressure continues to decrease rapidly up to the
point G. At this point, the exhaust-valve is fully open and the
pressiu'e decreases gradually as represented by the curve OA, to
the end of the stroke, where the pressure is practically atmos-
pheric, and the cycle has been completed. The press\u*e in the
cylinder, during expulsion, is higher than that of the atmosphere
due to the resistance offered by the exhaust-valve, to the outflow
of the products of combustion.
H
That part of the diagram, which represents the effects due to
throttling, has been purposely exaggerated.
The curve DE is usually, more or less, wavy; this may be
due to various causes. If the vibrations appear to be regular and
of decreasing amplitude, they are principally due to the inertia of
moving parts of the indicator. On the other hand, if the pressure
is apparently constant for a time, then suddenly decreases, etc.,
the waves are due to friction between the piston and cylinder of
the indicator. This trouble is readily removed by proper cleaning
and lubrication of the piston and cylinder. Furthermore, waves
may be established in the mixture in a manner similar to that
described in Art. 69; i.e., as the piston begins to compress the
IDEAL COEFFICIENT OF CONVERSION AND TESTS 217
mixture a wave of compression travels through the medium to
the other end of the cylinder, where it is reflected, with change of
sign. This reflected wave then travels toward the piston; and
when it meets the piston reflection again takes place, etc. In this
manner, inequalities in pressure may be established, which under
certain conditions may persist throughout the compression and
expansion strokes. However, in general, these inequalities will
not be manifested to any marked degree on the indicator diagram,
since the inertia, of the moving parts of the indicator, will tend
to suppress them.
By an inspection of Fig. 41, it is obvious that the work done
on the piston during the aspirating stroke is measured by the area
JABK; and, likewise, the work done by the piston during the com-
pression stroke is measured by the area KBCJ, Diuing combus-
tion, since there is no displacement of the piston, the work done
is zero. During expansion the work done on the piston is measured
by the area JDEFK. And, during expulsion, the work done by
the piston is measured by the area KFGAJ. By taking the
algebraic sum, we find that the net work done by the working sub-
m
stance, during the cycle, is measured by the difference between
the areas CDEFGi and AiB.
Hence, if the mean effective pressure is determined by means of
a planimeter, the tracing point of the planimeter, in tracing the
area AiB must travel in a sense opposite to that pursued in tracing
the area CDEFGi, That is, if i be the starting point, then, to
find the difference between the two areas, the tracing point of
the planimeter must follow, in order, the path i, C, JD, J5, F, (?,
I, A, B, i.
The area AiB represents the work lost, due to valve throttling,
and is, in well designed engines, small in comparison with the area
CDEFGi. If the power lost, due to valve throttling, is large in
comparison with the total indicated power, the valves must be
readjusted. In general, the spring which gives good results for
measuring the indicated power, has a modulus so high that the
218
THERMODYNAMICS
part of the diagram, representing the power lost, due to valve
throttling,, is too small to be accurately measured. But, by using
a stop, so as not to injure the spring, a much lower scale spring
may be employed. In this manner the power lost, due to throt-
tling, and also the amount of precompression may be accurately
determined.
167. Efficiency and Precompression. In Art. 161, equation
(5), it was shown from theoretical considerations that, other
things being equal, the thermal efficiency increases with the amoimt
of precompression. This is foimd to be so in practice. There
are, however, limits, beyond which the precompression may not
be carried, due to the severe strains to which the engine is sub-
jected diu'ing the explosion of the mixture.
Tests made, in the Cooper Union Laboratories, on a Fair-
banks 8 H.P. gas engine, gave the following results :
R.P.M.
B.H.P.
Efficiencien.
VjVi
Thermal,
Per Cent.
Mechanical,
Per Cent.
4.72
4.96
6.09
5.31
368
395
414
478
6.54
7.60
7.92
9.14
13.9
15.8
16.1
19.8
74.0
72.0
68.0
67.0
The value given for the B.H.P. is, in each case, the maximum
load the engine would carry for the given precompression. On
attempting to carry the precompression higher than that given
by t;i/v2 = 5.31, it was found that the vibrations set up in the engine
were so violent that satisfactory operation could not be obtained.
From the table it is seen that the thermal efficiency increases
rapidly with increased precompression. The mechanical efficiency,
however, is considerably reduced. The fuel used during these
tests was illuminating gas having a calorific value of about 590
B.T.U. per cubic foot. The amount of precompression which,
in any case, gives the best results depends, of course, upon the
quality of the fuel used.
IDEAL COEFFICIENT OF CONVERSION AND TESTS 219
It must, however, be emphasized that in any case, without
considering the severe strains to which the engine is subjected,
the amount of allowable precompression depends upon the tem-
perature of ignition for the fuel used. For, if the temperature of
the mixture due to the heat developed during the compression,
becomes higher than that of ignition, premature explosions will
occur, and the engine will not operate successfully.
168. T-(p Diagrams and Internal Combustion Engines. The
T- (p diagram, very frequently is a material aid in studying the
Fig. 42.
effect produced by a change in the cycle upon which an internal
combustion engine operates. As an example, if we plot the T- 9
diagram for a foiu'-phase cycle, the efifect produced by changing
the amount of precompression is obvious from an inspection of
the figure. Let A, in the T-? diagram. Fig. 42, represent the
condition of the mixture, as regards temperature and entropy,
at the end of the aspirating stroke. Then, since the compression
is assumed adiabatic, the entropy remains constant while the
temperature rises from T^, that before compression, to Ti, that
220 THERMODYNAMICS
after compression; the line representing this being parallel to
the T axis, and the condition of the mixture; as regards temperature
and entropy, is given by the point B. During combustion there
is a rise in temperature, and also, an increment in entropy. The
increment in entropy is given by
91
r^« dT
where M is the mass of the gas present, Cv the thermal capacity
per unit mass at constant volume, and T2 the temperature, when
complete combustion has taken place. Though the thermal
capapities of gases vary somewhat, for the ranges of temperature
obtaining in an internal combustion engine, the vanations are
probably not very large. Hence, so far as the present discussion
is concerned, no serious error is introduced by assuming C»
constant, and equation (6) becomes
9i=Maiog^ (7)
The curve BCj therefore, representing the relation of temper-
ature and entropy, during combustion, is logarithmic. During
the expansion, which is assiuned adiabatic, the entropy is con-
stant, while the temperature falls from T2 to Tz, Hence the
curve, CD, representing this change, is parallel to the T axis.
Finally, heat is rejected, the temperature falls from Ta to 74,
and the relation of change in temperature to change in entropy
is again logarithmic, as represented by the curve DA, By
assmning the thermal capacity of the products of combustion
constant, while the temperature falls from T^ to T4, we find for
the change in entropy,
92 = Maiog^ (8)
IDEAL COEFFICIENT OF CONVERSION AND TESTS 221
From Art 161 we have
hence, smce T2 is greater than Ti, T2 — Tz must be greater than
Ti — T^, and CD on the diagram, must be greater than BA,
Since, by Art. 109, the area under the curve, BC, is propor-
tional to the heat absorbed, and the area under the cm^e, AD^
is proportional to the heat rejected, it follows that the ideal coeflS-
cient of conversion is
^ Area FBCE-Areo, FADE _ Area ABCD . .
^ AresiFBCE Atqel FBCE' ' ' ^^
From equation (9), and by an inspection of Fig. 42, it is obvious
that the ideal coefficient of conversion increases with increased
precompression. Thus, if the precompression had been such that
the temperatiu'e at the end of the compression were T2, as repre-
sented by the point G, such that combustion takes place at the
constant temperature 7^2, the ideal coefficient of conversion would
be
Area AGCD
if) =
Area FGCE '
which is obviously greater than that specified by equation (9).
This is the principle upon which the Diesel motor operates; i.e.,
an attempt is made to bring about the application of heat at
constant temperature. Again, if after complete combustion
has taken place, the expansion be continued until the temperature,
as represented by the point K, has been reached, the ideal coef-
ficient of conversion is still further increased, and is given by the
relation
Area AGCK
TQ =
Area FGCE
which brings us back to the Camot cycle.
222 THERMODYNAMICS
But, as has been previously explained, in Art 115, this requires
a stroke of greater length than is consistent with economy.
169. Actual p-v and T-9 Diagrams of Internal Combustion
Engine. The quantity of heat which a gas absorbs, or liberates,
during a given temperature change, depends upon whether the
change takes place at constant pressure or at constant volume.
The change, however, usually takes place in a manner such that
neither the pressure nor the volume remains constant. When
both pressmre and volume vary, the change in entropy is readily
found from equations (42), (43), and (44), of Art. 49. Equation
(42) states that
dQ=^Cj4T~{Cp-C,)dp.
Assuming the process reversible, then, dividing through by T,
we have
and
from which
92-(pi = Cplog^-(C,-Qlog^^ . . . (10)
i 1 Pi
where <pi, Ti, and pi represent, respectively, the initial entropy,
temperature, and pressure, and 92, Tg, and p2 represent, respect-
ively, the final entropy, temperature, and pressure.
Equation (43), of Art, 49, states that
dQ = C^r+|(Cp-C,)dt;;
from which, by substituting for p/J? its value, T/y, and dividing
through by T, we find
dQ , ^dT . f^ ^ y.dv
— = d (p = C,-^+ (Cp- O— ;
IDEAL COEFFICIENT OF CONVERSION AND TESTS 223
and
from which
?2-fi = C.log^'+(C,-C,)log|^. . . . (11)
i 1 Vi
Again, equation (44), of Art. 49, states that
from which, by substituting for v/R and' p/R, respectively,
T/p and T/v, and dividing through by T, we find
and
from which
T p V
^9i -/pi P Jti v'
92-(pi = C.log2?+C;,log^ (12)
Equations (10), (11), and (12), then give the change in entropy,
respectively, in terms of the change in temperature and pressure,
the change in temperature and volume, and the change in pressiu*e
and volmne. Equation (12), however, is usually the most con-
venient; since, by means of a scale, after laying off on the indicator
diagram, the axis of zero pressure and zero volume, the pressures
and volmnes corresponding to various points of the diagram are
readily found; and from these, by choosing a suitable point for
zero entropy, the entropy corresponding to the various points
is easily computed. Dividing, equation (12), by C», we have
^(92-(Pi) = Iog2?+nlog^; .... (13)
224
THERMODYNAMICS
and since the scale employed, in plotting the r-9 diagram is
arbitrarily chosen, we may drop the factor 1/C» and employ the
equation
"... . (14)
(P2-9i = log^+nlog-.
Fig. 43 is a reproduction of an indicator diagram taken from
an 8 H.P. Fairbanks gas engine, operatmg on illuminating gas.
The ratio of the volume before compression to that after compres-
sion, was 5.31; and the scale of the spring used, in taking the
card, was 200 lbs. per square inch.
H
V
Fig. 43.
01 is the axis of zero pressure, obtained by measuring down,
to proper scale, from the atmospheric line, AA\ a distance repre-
senting the instantaneous barometric reading. OH is the axis
of zero volume, obtained by measuring, to proper scale, to the
left of the point A, a distance representing the equivalent length
of the clearance volume. AB represents the aspirating stroke,
SB' measures the rise in pressure, while the admission-valve is
closing, B'C represents the compression stroke, CD represents
the combustion, at practically constant volume, DE represents
the expansion of the products of combustion, EFG represents the
change from the time the exhaust-valve begins to open until
IDEAL COEFFICIENT OF CONVERSION AND TESTS 225
it is fully open, and GJ represents the remainder of the expul-
sion stroke. At the end of the expulsion stroke the exhaust-
valve closes, the admission-valve opens, the pressure falls from
/ to il, and the cycle is completed.
Fig. 44 is a representation of the T- 9 diagram, plotted from
the p-v diagram, as shown in Fig. 43. Assuming the temper-
ature at the point B, Fig. 43, to be equal to that of the atmosphere,
and employing the characteristic equation, the temperatures
corresponding to the various points were computed. An arbitrary
value of entropy for the substance, corresponding to the point
9
Fig. 44.
where the curves, representing compression and expulsion, inter-
sect, was assiuned. Then by substituting in equation (14), the
values of pressure and volimie, as found by scale, from the
p-y diagram, for the various points, the entropy, corresponding
to these points, was readily computed. The value of n employed
in making these computations, was a mean value, computed from
the constitution of the mixture, before combustion, for the com-
pression curve, and from the constitution of the exhaust gases,
for the expansion curve.
The point Aj Fig. 44, represents the condition of the working
substance, as regards temperature and entropy, at the point where
226 THERMODYNAMICS
the compression curve and expulsion curve intersect; and, the
curve AS represents the relation between temperature and entropy
from this point to the end of compression. BC is the curve, repre-
senting the relation of temperature and entropy, for combustion,
at practically constant volmne. CDE represents the relation of
temperature and entropy for expansion, and EA the relation of
temperature and entropy for the abstraction of heat at, practi-
cally, constant volume.
Though, as has been previously stated, the thermal capacities
of gases are not known for high temperatures, and therefore the
diagram does not rigidly represent exact conditions, it still appears
that, during compression, in this particular case, since the entropy
is increasing as the compression advances, the cylinder walls are
giving up heat to the mixture. From C to D the entropy of the
substance appears, from the diagram, to be increasing, which
is probably due to after burning; i.e., complete combustion has
not taken place when the power stroke begins. From D to E
the entropy first decreases and then increases, which appears
to indicate that for a time, while the temperature of the products
of combustion is high, heat is being given up by the gases to the
cylinder walls, and later, as the temperature falls, heat is abstracted
by the gases from the cylinder walls. In this manner, even
though the T- 9 diagram may not represent exact conditions,
conclusions may still be drawn, in regard to exchange of heat
between the cylinder walls and the working substance.
By assuming the compression adiabatic and computing the
temperature for the point B, a value was foimd which was con-
siderably less than that obtained, from Fig. 43, by means of the
characteristic equation. This also shows that, during compres-
sion, heat has been abstracted, by the gases, from the cylinder
walls.
170. Multi-cylinder Engines. Due to the fact that in a
four-phase cycle there is only one power stroke for every four
strokes, engines operating on this cycle are usually built with,
IDEAL COEFFICIENT OF CONVERSION AND TESTS 227
at least, two cylinders; but more freq\iently, especially for auto-
mobiles, with four, six, and sometimes, even with eight cylinders.
The explosions are so timed that the turning moment on the
shaft throughout a rotation of the fly-wheel is, as nearly as pos-
sible, uniform.
171. Double-acting Cylinders. For power plants, internal
combustion engines are, at the present time, frequently con-
structed so as to be double acting; i.e., explosions properly
timed, are brought about in both ends of the cylinder. However,
for a well distributed thrust there must be two such cylinders,
which may operate either tandem or twin.
172. General Outline of Test. As stated in Art. 167, actual
tests show that the thermal efficiency of an internal combustion
engine is a fimction of the amount of precompression. Hence,
in making a complete test of an engine, it is necessary to determine
what precompression yields the best results. To do this, various
methods may be employed; one, conveniently carried out, is
that of fastening, by means of machine screws, disks of various
thicknesses, and diameters equal to that of the cylinder bore, to
the end of the piston, thus changing the clearance. In general,
it will be found that, for every change in the amount of precom-
pression, an adjustment of the governor is necessitated, so that
the load is properly carried. At the same time, the calorific value
of the fuel is determined, as well as the chemical constitution of
the exhaust gases. If, by a chemical analysis, it is foimd that the
exhaust gases show incomplete combustion, the amount of air
admitted, during the aspirating stroke, must be changed, by a
change in the admission valves, until an analysis of the exhaust
gases shows complete combustion. Proceeding in this manner,
step by step, it will be foimd that for every engine, there is a
definite precompression which yields a maximal thermal efficiency;
and, when this has been determined, the various losses are
readily found.
To determine the various losses, the load on the engine is
228 THERMODYNAMICS
maintained constant, for a given run, and noted. This gives the
output of the engine. From the quantity of fuel consumed,
samples of which are tested for calorific values, from time to time,
during the run, the total energy consumed is found. The differ-
ence between energy consumed, during the run, and the work
delivered by the engine, during the same interval of time, con-
stitutes the combined losses. The losses are chiefly: Mechanical
losses, heat carried away by exhaust gases, heat carried away
by the cooling water, heat lost by incomplete combustion, and
heat lost, from the surface of the engine, by radiation and con-
vection.
The mechanical losses are determined by taking the difference
between the indicated work and the work delivered. It is, of
course, necessary to take a nmnber of indicator diagrams, during
the run, so as to obtain an average value for the indicated work;
and furthermore, the number of power strokes should be deter-
mined from the number of explosions, rather than from the num-
ber of rotations, made by the fly-wheel.
To determine the heat, carried away by the exhaust gases,
the temperature of the gases, at the exhaust port, is determined
by means of a pyrometer. And, from the constitution of the
exhaust gases, as foimd by analysis, the mean thermal capacity
per imit mass is found. Again, from the constitution of the fuel,
the mass of fuel consumed, and the constitution of the exhaust
gases, the total mass of the exhaust gases is readily computed.
Finally, by taking the product of the difference in temperature
between the exhaust gases and the room, the mass of the exhaust
gases, and the mean thermal capacity per unit mass, the heat
carried away by the exhaust gases is foimd. It must, however,
be remembered that the thermal capacities for gases, as given in
tables, are, in general, the results obtained by experiments con-
ducted between the limits of 0°C. and 200°C.; and, it is not at
all certain that these values are correct for high temperatures.
The heat carried away by the jacket water is determined
IDEAL COEFFICIENT OF CONVERSION AND TESTS 229
directly from the difference in temperature, between inflow and
outflow, and the mass of water flowing through the jacket during
the run. The temperatm'es are foimd by means of ordinary
thermometers; and, the mass of water, by collecting in a suitable
vessel and weighing.
The heat lost, due to incomplete combustion, which should
be very small, is computed from the constitution of the products
of combustion.
There finally remains, then, the heat lost by radiation and
convection. This cannot be found directly, but is assmned to
be equal to the difference between the input and the sum of the
other losses plus energy delivered.
Having found the various losses, a comparison may be made
between the engine under test and other engines; and conclu-
sions drawn therefrom in regard to making changes in the design
or the method of operation.
CHAPTER XVI
COMPRESSED AIR AND COMPRESSORS
173. Compressed air is used extensively and for a variety of
purposes. It is used in tunneling; in mining, where, after it
has done work upon an air motor, it may be employed for venti-
lating purposes; in general, for power transmission; for air
brakes on trains; etc. Hence, it is important that the compres-
sion be brought about in the most economical manner possible.
To do this, in attempting the design of an efficient air compressor,
it is necessary to consider the imderlying principles of thermo-
dynamics, as well as those of machine design.
However, before developing the formula representing the least
amount of work that must be done in compressing a given mass of
air, from one pressure to another, we will, as a matter of conve-
nience, first determine the constants for air, and discuss briefly
isothermal and adiabatic compression.
174. Air Constants. The density of air, under a pressure of
one standard atmosphere (1.01325X10® dynes per square centi-
meter) and at a temperature of 32°F., is 0.001293 grams per
cubic centimeter,' Converting this to pounds and ciJbic fedj we
find for the density, 0.08072 pounds per cubic foot; and, from this,
a volume of 12.39 cu.ft. per pound. The pressure of one stand-
ard atmosphere, expressed in gravitational units, is, approx-
imately, 2115 lbs, per square foot; and, this corresponds, very
nearly, with a pressure of 14.7 lbs. per square inch. Taking the
product of pressiu-e and volimie, we find
pt; = 2115X12.39=26,200 lbs. per sq,ftXcu.ft
230
COMPRESSED AIR AND COMPRESSORS 231
Since 32°F. corresponds, approximately, to 491.6 on the ther-
modynamic scale, we find, from the characteristic equation, for
1 pomid of air,
^ 491.6 -^^•^•
The foregoing constants have all been given to the nearest figure
in the fourth place; but, in general, results of suflicient accuracy
will be found by rounding off to the nearest figure in the third
place. For, in designing compressors, some assumption has to be
made regarding the average annual temperature of the atmos-
phere; and, frequently, it is assimied that this average temperature
is 62°F. This, however, is not the proper value for all cases and
localities. Assmning 62®F. to be the proper temperature to be
employed, then we shall have, for the product of pressure and
volume,
PaVa =RTa= 53.3 X 522 = 27,800 ;
where Ta is the temperature on the thermodynamic scale, corre-
sponding to 62°F., Pa is the atmospheric pressure, and Va the
corresponding volume, for 1 poimd of air at 62 °F.
176. Compression and Expansion. Assmne the purpose,
for which the compressed air is employed, to be that of driving
an air motor; the construction of an air motor, and the cycle
upon which it operates, being very nearly the same as that of a
steam engine. Let it be further assumed that the compression,
in the air compressor, takes place in a manner such that the heat
developed, by the compression, is immediately absorbed by the
surroundings; i.e., the compression is isothermal. The intrinsic
energy of the air, then, at the end of the process, is precisely the
same as at the beginning. Again, if it be assumed that the
expansion in the air motor takes place in a manner such that the
heat required during the expansion, in doing external work, is
unmediately supplied from the surroundings, as required, then the
232 THERMODYNAMICS
intrinsic energy of the air, at the end of the process, is precisely
the same as at the beginning. And under these assumed condi-
tions, the work done on the air, while being compressed, is pre-
cisely equal to the work done by it, while expanding, between the
same limits of pressure. It is thus seen that a vessel, containing
compressed air, at room temperature, does not constitute a reser-
voir of energy; but, the air is merely in a condition such that it
can absorb energy, in the form of heat, from the surroundings,
and convert it into mechanical work. On the other hand, if the
compression is adiabatic and the vessel, in which the compressed
air is stored, is perfectly insulated, such that the heat developed
during compression is retained, then the vessel does constitute
a reservoir of energy; for, by an adiabatic expansion, the work
done on the air, during compression, may again be recovered; and
the air at the end of the expansion is in precisely the same con-
dition as it was at the beginning, without any exchange having
taken place between the air and the surroundings. This, of course,
assumes no other losses. The receiver, then, in the latter case,
containing the compressed air, constitutes a reservoir of energy,
not because it contains compressed air, but merely because heat
has been stored which may be reconverted into mechanical work.
In general, however, when air is compressed, the compression
is very nearly adiabatic; and the air is stored in a receiver,
where, in a very short interval of time, the heat developed
during compression is, by means of conduction and radiation,
given up to the surroundings, and is irrevocably lost. Hence,
when expansion takes place in the motor, which is again, approx-
imately, adiabatic, the external work is done at the expense
of the intrinsic energy of the air; and, accordingly, the tem-
perature falls. Hence, since the intrinsic energy of the air is
practically a function of the temperature only, it follows that
the intrinsic energj'^ at the end of the expansion, in the motor,
is less than it was mitially in the compressor, at the instant
when compression began.
COMPRESSED AIR AND COMPRESSORS
233
176. Isofhermal Compression and Expansion. The results
just deduced, in Art, 175, are best illustrated by means of the
ideal p-v diagram; first, by considering isothermal processes, and
then by considering adiabatic processes. Let, in Fig. 45, 01
and OH be, respectively, the axes of zero pressure*^ and z^ro
volume; and, for the present discussion, it will be assimied that
the compressor has no clearance. The work done, then, on
the piston hy the air, during the aspirating stroke, as represented
by the Une AB, is measured by the area OABE, During the
isothermal compression, represented by the curve EC, the work
done by the piston, on the air, is measured by the area EBCF.
At the point C, the air is imder a pressiu*e equal to that obtaining
in the receiver, the exhaust-valve opens, and the air under a
constant pressure, as represented by the line CD, is forced into
the receiver, and the cycle is completed. The work done by
the piston on the air during expulsion, is measured by the area
FCDO. Finally, the net work done by the compressor, during
the cycle, is ob^viously measured by the difference between the
areas EBCDO and EBAO; i.e., the area ABCD.
Representing the pressm-e and voliune of the air, corresponding
to the point B, respectively, by pa and Va, and likewise, for the
point C, by pi and vi, then, by considering the work done on
234 THERMODYNAMICS
the piston negative, and the work done by the piston positive,
we have for the work done, during the aspirating stroke,
. Wi = -paVa (1)
The work done, during the compression, BC, is
1^2=- Cpd^l (2)
and the work done, during expulsion, is
Wz=pivi (3)
Finally, the net work done, during the cycle, is the algebraic
of TTi, W2, and W3; i.e.,
£1
pdv+pivi; (4)
where W is the net work done. But, since the compression is
isothermal,
PaVa = pivi ;
m
and equation (4) becomes
W^-jpdv (5)
Substituting now, in equation (5), for dv its value, as found from
the equation
pV = PaVa,
we find
— = PaVa log — (6)
Pa P Pa
If we are dealing with 1 pound of air, and assimie the tem-
perature, during the aspirating stroke, to be 62°F., then we may
COMPRESSED AIR AND COMPRESSORS 235
substitute, in equation (6), the value of paVa as found in Art.
174, and we find
W = 27,800 log 21 f t.-lbs. per pound. ... (7)
Pa
That is, the right-hand member of equation (7) is the expres-
sion for the amoimt of work in ft.-lbs., that must be done in taking
in, under the assumed conditions, 1 pound of air at a pressure
Pa, compressing it to a pressure pi, and forcing it against this
pressure into a receiver. It is obvious that the ratio pi/pa is
independent of the unit of measure chosen; but, it is necessary, in
order to obtain the work in ft.-lbs., that the nmnerical coefficient,
in equation (7), be deduced by expressing the pressure in lbs. per
square foot and the volume in cubic feet.
Assume, now, that the process takes place in the reverse
order, step by step, in a manner such that the diagram is traced
in the order, DCBA. By assuming a perfect regenerator, such
as was discussed in Art. 27, the process becomes ideally reversible,
and the net work done, on the motor, is precisely equal in
amount to that done by the compressor, as given by equation
(6). And since this is the best that can be done in any case,
the indicator diagram, representing isothermal compression, is
the one chosen as a standard, for the comparison of the per-
formance of air compressors.
177. Adiabatic Compression. For the same initial and final
pressiu-es, the work done, during compression, and also during
expulsion, will not be the same for an adiabatic process as it is
for an isothermal process. On the other hand, the work done
during the aspirating stroke is the same in either case.
Let in Fig. 46, the Une AB represent the aspirating stroke,
the curve BE the adiabatic compression, and the line ED the
expulsion stroke at the pressure of the receiver. The net work
done by the piston is measured by the area ABED. Had the
compression been along the isotherm BC, the net work done
236
THERMODYNAMICS
would be measured by the area ABCD; hence, the excess of work
done, during a cycle, when the compression is adiabatic, over
that done when the compression is isothermal, is measured by
the area BCE.
The work done during the aspirating stroke is again
Wi==—paVa;
(8)
where the symbols have the same significance as in Art 176.
The work done, during adiabatic compression, is
W2
J 'Pi
pdv]
Pa
(9)
but, since we are now dealing with an adiabatic change, the
value of dv to be substituted in equation (9) must be deduced
from the equation
Solving for v we find
from which.
dv=
1 _i
V^Pa^'VaP ^
1 1 1-Hi
Pa" VaP *» dp.
n
COMPRESSED AIR AND COMPRESSORS 237
Substituting in equation (9), we find
n-\ n-1
= ^^^(P1 n -p„ n )
n-1
_ PaVg
n-1
Ite)"-
1 (10)
Since now the volume, at the end of the adiabatic compression, is
the work done, during expulsion, is
n-1
Ws^piVi^Pa'^VaPl »» =PaVa{—j • • (H)
Taking the algebraic sum of the right-hand members of equa-
tions (8), (10), and (11), we find, for the net work done by the
compressor,
which reduces to
n-1
. . . (12)
-—,PavJ(^
W '^VaVa\l^] ^ -1
It was stated, in Art. 176, that the cycle having isothermal
compression may be taken as a standard cycle. The ratio of
the work done, during a cycle, when the compression is iso-
thermal, to that when the compression is adiabatic, may be
termed the theoretical efficiency of compression. Dividing equation
238
THERMODYNAMICS
(6) by equation (12), we find, for the theoretical eflBiciency of
compression.
^ =
log^i
Pa
(13)
As a matter of convenience a table is here given, which was
obtained by computing the theoretical eflBiciencies, by means of
equation (13), on the assumption that n has the value of 1.4.
Pi
Pa
Per Cent.
Pi
Va
Per Cent.
1.5
94.3
6
76.6
2
90.4
7
74.8
3
85.1
8
73.2
4
81.5
9
71.9
5
78.8
10
70.7
•
It will be noted that, when the ratio pi/pa=3, the theoretical
eflSciency of compression is approximately 85 per cent, and when
pi/pa=4, it is approximately 81.5 per cent. Hence, when the
ratio of pi to pa is greater than 3 or 4, the losses, from a ther-
modynamic standpoint, become excessive. Therefore, if it be
desired to operate economically, it becomes necessary to limit
the ratio of final to initial pressure to a value between 3 and 4.
Equation (12) may be transformed so as to express the work
done by the compressor, in terms of the initial and final tem-
peratures. Since
n-l
Ta \Pa)
where Ta is the initial and Ti the final temperature, and since
PaVa^RTaf
COMPRESSED AIR AND COMPRESSORS 239
we find, by substituting in equation (12),
And finally, by substituting, in equation (14), for n and R,
respectively, the equivalent values, Cp/Cv, and J(Cp— Ct), we
find
W=JCp(Ti-Ta) (15)
That is, equation (15) shows that the net work consumed by an
air compressor, per cycle, per pound of air, when the com-
pression is adiabatic, is precisely equal in amount to the heat,
expressed in mechanical units, consmned in elevating the temper-
ature of the air, at constant pressure, from that before compres-
sion to that after the compression is completed.
Equation (15) may be deduced in an entirely different, though
very simple, manner. The work done by the piston, during
admission, is
And, during adiabatic compression, since the intrinsic energy of
the air is a function of its temperature only, the work done by
the piston is
W2=C,(Ti-Ta)J'. (jb):f^^£rydt4.
The work done by the piston, during exhaust, is
W3 = piVi=RTi=JiCp-C,)Ti; (c)
taking the sum of TTi, 1^2, and TFa, as given by equations (a),
(6), and (c), we find
W=JCpiTi-Ta).
178. Multi-stage Compression. Various methods, in which
an attempt is made, to bring about compression, approaching
240 THERMODYNAMICS
an isothermal process, have been tried; but, it appears impossible
to reduce the exponent, n, to a value approaching imity. The
various methods used are: Water-jacketing, playing a jet of
water into the cylinder while compression is taking place, and
spraying, by means of an atomized jet of cold water, the air while
it is being compressed. Water-jacketing appears, so far as the
results of investigations show, to give very little, if any gain in
economy. That is, when recourse is had to jacket cooling, the
heat developed by compression is absorbed so slowly that the com-
pression is practically adiabatic. When the cooling is attempted
by means of a jet of water, played into the cylinder, the exponent,
n, may be reduced to a value of about 1.35; and when the
cooling is brought about by an atomized spray, the value of n
may be reduced to about 1.25. Hence, at best, the compression
is far from approaching an isothermal process; and, for eflBicient
operation, when the ratio of the final pressure to the initial
pressure is greater than 4, recourse must be had to midti'^tage
compression.
We will consider, first, a two-stage compressor. That is,
during the aspirating stroke, a certain quantity of air, at a
pressure pa, flows into the cylinder, which on the return stroke
is compressed to some pressure pi; the relation of pressure
and volume being given by the equation
pv^ = k;
where the exponent n, depending upon the method of cooling
applied, may have a value ranging from about 1.25 to 1.4. At
the end of this compression stroke, when the pressure pi has
been attained, the air is expelled into a receiver imder a con-
stant pressure pi. The receiver has a jacket through which
water is circulated in a manner such that the heat developed,
during compression, is removed; and the product of pressure
and volume, after cooling in the receiver, is equal to the product
of pressure and volume at the instant the compression began.
COMPRESSED AIR AND COMPRESSORS
241
In other words, the condition of the air, as regards pressure and
volume, in the receiver, is the same as though the compression
had been isothermal; i.e.,
Let, in Fig. 47, the line AB represent the aspirating stroke,
for the low-pressure cylinder, and the pressure and volimie,
corresponding to the point B, be given, respectively, by pa and
Va. The compression then takes place approximately adiabat-
FiG. 47.
ically, as represented by the curve BE. At the point E the
exhaust-valve opens and expulsion takes place, to the receiver,
at a constant pressure pi. The volume of the receiver being large
in comparison with that of the first cylinder, the pressure in it
is sensibly constant; and the volim^ie of air, in cooling from Ti,
the temperature at the end of the compression BE, to Ta, the
atmospheric temperature, shrinks from the volume, represented
by FEj to that represented by FG, such that
and the point G is on the isotherm through the point B, For the
same quantity of air, then, the line FG represents the aspirating
242 THERMODYNAMICS
stroke for the second, or high-pressure, cylinder. The curve GJ,
represents the compression, which is practically adiabatic, to the
pressure p2, existing m the reservok in which the air is stored.
Finally, the expulsion stroke to the reservoir is represented by the
line JD.
The various quantities of work involved during the cycle are
as follows: During the aspirating stroke, for the low-pressure
cylinder, the work done en the piston is measured by the area
ABKO; and during compression, to the pressure pi, the work
done by the piston is measiured by the area KBEL; and during
expulsion, to the receiver, the work done by the piston is measured
by the area LEFO, Hence, the net work done by the piston, in
the low-pressure cylinder is measured by the area ABEF, In a
similar manner, we find that the net work done, by the piston in
the high-pressure cylinder, is measured by the area FGJD. The
total net work done, therefore, by the two-stage compressor,
during the cycle, is measured by the area ABEGJD. Had the
compression taken place in a single-stage compressor, between the
same limits of pressure, the net work done, by the piston, would
be measiu'ed by the area ABEND; where the curve BEN repre-
sents an adiabatic through the point B. Hence, the saving in
work, neglecting losses, by employing a two-stage compressor, is
measured by the area ENJG, And the work done by the piston
of this two-stage compressor, in excess of that which would have
been done had the compression been isothermal, is measiu*ed by
the sum of the areas GBE and CGJ, where the curve BGC repre-
sents an isotherm.
By equation (12), we have for the work done in the low-pres-
sure cylinder, per cycle, when 1 pound of air is taken in at a pres-
sure Pa, is compressed adiabatically to a pressure pi, and expelled
at this pressure to a receiver,
n-l
Wl = -^aVal l^) ' -ll (16)
COMPRESSED AIR AND COMPRESSORS
243
In a similar manner, the work done in the high-pressure cylinder,
when 1 pound of air is taken in at a pressiu*e pi, is compressed
adiabatically to a pressure p2, and expelled at this pressure to
a receiver, is
n-l
Since, however, the air in the intermediate receiver has its tem-
perature reduced to the initial value, we must have
PlVl=PaVa;
and equation (17) becomes
1^2 =
n-l
n { (V2\ ^
(18)
Taking the sum of equations (16) and (18), we find that the total
work done, during a cycle, by the two-stage compressor, is
^-^'^^'-~iv4(^f ^($y -A
(19)
Now, Pa is a constant and so is j>2\ since, for any particular
case, p2 is the desired final pressure, pi, however, is a variable;
and the value of W obviously depends upon the value chosen for
pi. Since, the only variable in the right-hand member of equation
(19) is pi, the value found for TF is a minimum when the expression,
is a minimum. Differentiating this expression, with respect to
pi, equating to zero, and solving for pi, we find
Pi = Pa*P2*.
(20)
244
THERIVIODYNAMICS
Substituting the value of pi as given by equation (20), in equations
(16) and (18), we find
and.
n-l
--;^-|fer-'l
(22)
From equations (21) and (22) it is seen that, if the work done,
during a cycle, by a two-stage compressor, is to be a minimum,
it must be equally divided between the two cylinders. Taking
the sum of the right-hand members of equations (21) and (22),
we find the net work done, when employing the most efficient
compression possible, by a two-stage compressor, in taking air
under a pressure pa and expelling to a receiver, under a pressure
P2, IS
w 2n
V2
n-l
2n
-1
ft.-lbs. per pound.
(23)
If the compression is brought about by three stages, the final
pressure being pa, and the pressures of the intermediate receivers,
respectively, pi and p2, then, on the assumption that, in the two
intermediate receivers, the temperature is reduced to that of the
atmosphere, the work done in the first, second, and third cylinders
is given, respectively, by
Wi =
n
TF2 =
n-l
_n_
n-l
(24)
. . (25)
and
n-l
w^-;hH^^ ' -'
(26)
COMPRESSED AIR AND COMPRESSORS
245
Taking the sum of Wi, W2, and Wz we find, for a cycle, the net
work done by the three-stage compressor, is
The right-hand member of equation (27) is a minimum when the
expression included in the brace is a minimum. Differentiating
this expression, first, assimiing pi variable, and pa, P2, and ps
constant, equating to zero, and solving for pi, we find
pi = Vp^2 (28)
Equation (28) gives the relation of pi to pa and p2 such that the
process in going from pa to p2 shall involve a minimum amoimt
of work. Differentiating again, this time, however, assuming
Pi and p3 constant, and p2 variable, in order to obtain the relation
P2 must bear to pi and pz such that a minimum amount of work
is involved, while the process takes place from pi to pz, we find
P2 = Vpipz
(29)
By elimination we find, from equations (28) and (29),
and
Pi = ^Pa^PZj
P2 = i^
VaPz'
(30)
(31)
Substituting, in equations (24), (25), and (26), the values of pi
and p2, as given by equations (30) and (31), we find
'^-»-v-£
n-l
3n
-1
, • • • • \<ij^)
n-l
^"ir^-m"-'
',.... (33)
and
n-l
Wz^
n
^•"•1©"-';
(34)
246 THERMODYNAMICS
Equations (32), (33), and (34), again show that, for the most
economical compression, the work must be equally divided between
the cylinders; and the net work, for the three-stage compressor, is
n-l
W=~^PaVa (— ) ^'^ -1 ft.-lbs. per pound. . (35)
In a similar manner, for the most economical four-stage com-
pressor, we find
n— 1
4/1 / 7^4 I ***
W = zPaVa ( ^ ) — 1 ft.-lbs. per pound.
By a comparison of equations (12), (23), (35), and (36), it is
seen|that in each case the coefficient n/{n— 1), is multiplied by the
number of stages, and the exponent (n— l)/7i, is divided by the
number of stages; and in all other respects, the equations are
identical. We may then write a general equation for the net
work done by a naulti-stage compressor, in taking in 1 pound of air
under a pressure pa, compressing it, by means of S stages, to a
pressure p, and expelling it, at this pressure to a reservoir. This
net work done is given by
n-l
W = TPaVaU—j -1
^-1 [\Pa/
ft.-lbs. per pound. . (37)
It is obvious that for any two given initial and final pressures,
the greater the number of stages, the smaller the areas, representing
the difference in work, between adiabatic and isothermal compres-
sion, become; and hence, the nearer the compression approaches
an isothermal process. And, in the limit, as jS in equation (37)
becomes indefinitely large, the compression becomes isothermal.
However, increasing the number of cylinders, increases the biilk
and first cost of the compressor, as well as the loss of work due
to friction and imperfect valve action. On the other hand there
are also certain mechanical advantages in a multi-stage com-
pressor, similar to those discussed, in Arts. 141 and 144, for the
COMPRESSED AIR AND COMPRESSORS
247
compound engine. That is, for a two-stage compressor the
stresses in the moving parts are less than in a single-stage com-
pressor operating between the same limits of pressm-e; and, if
the compressor be cross-compound, the thrust on the crank bear-
ings is more uniform. But, in any given case, there is a limit-
ing value for the niunber of stages; practically when the thermo-
djniamic gain is offset by the interest on the extra capital invested,
plus depreciation and mechanical losses. For final pressures
of about six atmospheres, twoHstage compressors are usually
employed.
179. Clearance. The clearance of an air compressor is very
small in comparison with the clearance of a steam cylinder. Still,
account must be taken of the clearance in designing a compressor;
since, due to it, the effective volumetric displacement of the piston
is less than the actual volimietric displacement. Let, in Fig.
48, AB represent the displacement of the piston, BC the com-
pression, CD the expulsion, and EA the clearance. Then, since
the volume of SkiifEA, remaining in the cylinder, when the exhaust-
valve closes, is under a pressure pi, expansion must take place to
atmospheric pressure, as represented by the curve DF, before the
inlet-valve opens and a new supply of air flows into the cylinder.
Hence, the effective piston displacement is measured by FB.
248 THERMODYNAMICS
Let K* he the ratio of the actual piston displacement, AB^
to the clearance, EA^ then
AB
For the expansion along the curve JDF, we may write
ViVi'^VavJ') (39)
where vi and Va are the volumes, respectively, as represented by
EA and EF. From equation (39), we find
i_
and, by substituting for v\ and Va, respectively, the value of EAj
as given by equation (38), and EF^ we obtain
Now, the effective displacement is given by
FB=AB-{EF-EA); (41)
hence by substituting, in equation (41), the value of EF, as given
by equation (40), and the value of EA, as given by equation (38),
we find
FB=AB
H(ff+h}- :■■- «^>
The effective piston displacement not being equal to the actual
piston displacement, does not affect the expression deduced for
the work done on an air compressor; for, the air remaining in the
cylinder, at the end of the expulsion stroke, does an amount of work
* K usually has a value of about 50.
COMPEESSED AIR AND COMPRESSORS 249
on the piston, in expanding, which is practically equal to that
which was done on it while being compressed. The efifect of the
clearance, then, is merely to reduce the capacity of the cylinder.
180. Throttling and Other Imperfections. The capacity of a
cylinder of an air compressor is very frequently more seriously
affected by other causes than it is by clearance. In the first place
there is always, due to valve friction, a certain amount of throt-
tling, which causes the pressure in the cylinder, during the aspira-
ting stroke, to be less than atmospheric. Further, due to imper-
fect valve action, i.e., the valves not opening or closing at the proper
time, the capacity is reduced. And, finally, the temperature of
the cylinder walls is usually higher than that of the incoming air,
which again tends to reduce the capacity of the cylinder. These
combined causes may reduce the apparent capacity, depending
upon the speed of the machine, from 5 to 20 per cent.
181. Adiabatic Expansion in Motor. The cycle of an air
motor is practically the reverse of that of an air compressor. The
admission-valve opens and air from the mains, under a practically
constant pressiu-e, forces the piston forward to the point of cut-
off, and the work done on the piston, per pound of air is
Wi = pivi; (43)
where pi is the pressure in the main, and vi the volume of one
pound of air at cut-off. The expansion is then practically adia-
batic, and the work done in expanding from the pressure pi, to
Pa, that of the atmosphere, is
n-l
The exhaust-valve then opens, the air is expelled imder a pressure
Pa, and the work done, by the air, is
L n-l
Wz= —PaVa— —pV^ViPa » (45)
260
THERMODYNAMICS
Taking the sum of the right-hand members of equations (43),
(44), and (45), we find, for the net work done on the motor,
per poimd of air,
n-l
n-l
n-l
n
n-l
plVi \ 1
-fe)"
ft.-lbs.
(46)
Since the temperature in the mains is practically atmospheric,
piVi is the product of pressure and volume, for one pound of air
under ordinary conditions, and may be replaced by the constant
27,800. Hence, equation (46) becomes
Tr=27,800-^|l-(^)""
n-i[ \pi/
(47)
The indicator diagram for the preceding discussion is shown
in Fig. 49, in which AB represents the admission, BC the expansion.
V
Fig. 49.
and CD the expulsion. The work done on the piston during
admission and expansion, is measured by the area ABCOF; and
the work done by the piston, during exhaust, is measured by the
area CGFD. Hence, the net work done, by the air, is measured
by the area A BCD.
COMPRESSED AIR AND COMPRESSORS 251
Equation (46) may be put into another form, by substituting
n-l
for pivi, its value RTi, and for {jpJvi) " t its value Ta /Ti.
Making these substitutions, we find
'^'^h'^H'-r)'
from which, for adiabatic processes, since in that case n=Cp/Cv,
we obtain
And, since
we have finally
72 = J\Cp — Cv) y
W^JCp{Ti-Ta) (48)
It must be noted that, in equation (48), Ti is the temperature in
the mains, which is practically that of the atmosphere, and Ta,
the temperature of the air after expanding adiabatically from the
pressure pi, that existing in the mains, to pa, that of the atmos-
phere.
182. Reheating. When air at atmospheric temperature,
and under a high pressure pi, expands to atmospheric pressure pa,
the corresponding temperature, Ta, will be very low. As an exam-
ple, if air under a pressure of five atmospheres, and at atmospheric
temperature Ti, expands adiabatically to a pressure of one atmos-
phere, its temperature becomes approximately,
n-l
Ta = Ti(^) "" =5^^(iy =330=-130°F.;
n having been assumed to have the value 1.4. Temperatures
as low as this, due to the fact that the moisture present in the air
freezes, makes lubrication difficult and clogs the valves, are
undesirable at the exhaust of an air motor. To prevent this,
252 THERMODYNAMICS
recourse must be had to reheating; i.e., the air from the mains
is passed through a heater before being admitted to the motor.
In being heated at constant pressure, the volume of the air
is increased, and the ratio of the two volumes is given by T/Ti)
where T is the temperature of the air after heating. Hence,
equation (46), giving the work done, per pound of air, on the air
motor, becomes
'^'-i;x;^^'"'{'-fer} (*»>
If, now, Ta is the temperature at the end of the adiabatic expansion,
to the pressiu*e Pa, and since n — Cp/Cvy piVi=RTi =J{C^—C^Ti,
n-l
and {pa/pi)~^ = Ta/Tiy we find, by substituting in equation (49),
TFi=|]x^^J(C,-c.)ri(i-^'). . .
(50)
And further, since the ratio of final to initial pressure is the same
whether there be reheating or not, the ratio of final to initial
temperature must also be the same for both cases; hence,
T
a ~ ■* /T7 I
where Ta is the finaL temperature when there is no reheating.
Substituting this value of Ta in equation (50), and simpUfying,
we find
W^^JC,^{T^^Ta) (51)
Equation (51) is the expression, in terms of the three temper-
atures, with reheating, for the work done per poimd of air, on the
air motor. Equation (48) is the expr^sion for the work done per
poimd of air without reheating. Taking the difference between
equations (51) and (48), we find, due to reheating, for the gain
in work
1^' - JCvY^Tx - Ta) -JCj^iTi - Ta) = JC^iTi - Ta)~~^. (52)
COMPRESSED AIR AND COMPRESSORS 263
The heat consumed, expressed in mechanical imits, in raising
the temperature of 1 pound of air at constant pressure, from Ti to
r,is
jCp(r-ri);
and the work which could be realized from this quantity of heat,
by means of a Camot cycle, is
Tr"=JCp(r-ri)?^ (53)
Taking the ratio of TT', as given by equation (52), to T7", as given
by equation (53), we find
In equation (54), the first factor, viz, T/Ti, is always greater
than unity, and for any given case, T\ — Ta is a constant. Hence
the ratio, W'/W', is greater than unity until the air is reheated
to a temperature such that
T T-Ti
Ti Ti-T,
(55)
And, for reheating to a temperature higher than this, the ratio
becomes less than unity. Solving equation (55), for T, we find
JT 2
r=^ (56)
Hence, for reheating to temperatures lower than that given by
equation (56), there is a thermodynamic gain; i.e., the gain in
work, due to the heat applied in reheating the air, is greater than
that which could be realized if an equal quantity of heat were
utilized on a Camot cycle for the same limits of temperature.
To illustrate, we will assmne a particular case and solve for
T. Let pi, the pressure in the mains, be six atmospheres, Ti be
522, and pa, the final pressure, be one atmosphere; then
254 THERMODYNAMICS
From equation (66), we find
r.^'-<W^.87i+.
522
w
Giving a temperature for reheating, above that existing in the
mains, of practically 349° F., which is higher than ever employed
in practice.
The heat consmned in reheating is applied to much better
advantage than in the case of a steam engine and boiler. Further-
more, since the fuel used in reheaters may be of a much lower
grade than that ordinarily employed for heat motors, there is a
saving in cost of fuel.
183. Loss of Head in Transmission Pipes. When a liquid
flows in a pipe, there is always, due to friction between the liquid
and the surfaces with which it comes into contact, a resistance
to be overcome; and on account of this, there is a loss in pressure.
That is, when friction is taken into account, BernouilWs Theorem,
which states that, for the steady flow of a liquid in parallel stream
lines without friction, the pressure head plus the velocity head plus
the static head is a constant for any section under consideration,
no longer applies. The energy consumed in overcoming fric-
tion, is manifested by eddy currents. These eddy currents in
tiu'n subside; and the energy, possessed by them, is converted into
heat, which in turn is lost by being dissipated to the surroundings.
The loss of energy thus experienced by a given mass of the liquid
is usually expressed by a loss of head. That is, the loss of head,
experienced by a unit mass of the liquid, is numerically equal to
the vertical height through which it would have to fall to do an
amount of work equal to that consiuned in overcoming the
friction.
From a great number of experiments upon the flow of liquids
in pipes, the following facts have been adduced: The loss of head
is very nearly proportional to the square of the speed of flow,
COMPRESSED AIR AND COMPRESSORS 255
varies directly as the length of the pipe, as the wetted perimeter,
and inversely as the cross-sectional area of the stream.
These relations may be stated symbolically as follows:
^=/Jr' (57>
where / is an experimental constant depending upon the nature of
the liquid and inner surface of pipe, and H, s, L, P, and A are,
respectively, the loss in head, speed of flow, length of pipe, wetted
perimeter, and area of stream. For any particular cross-section
the ratio of -4. to P is a constant; which is termed the hydraulic
radius, and may be replaced by the symbol K. Hence, equation
(57) may be written
Since the temperature of air, flowing in a pipe of any consider-
able length, is sensibly constant, the product of pressure and volume
is also practically constant; and hence, as the pressure falls the
speed of flow must increase. Therefore, since equation (58)
assumes a constant speed, it is not directly applicable to the flow
of air, or any other gas. In the limit, however, we have
dH=f^dL (59)
And, since the loss of head is numerically equal to the work done
by a unit mass of the substance, we have
dH = pdv; (60)
where p is the pressure, and dv the change in volume, per unit
mass, for the section under consideration. From equations
(59) and (60), we find
pdv=f£^dL (61)
256 THERMODYNAMICS
There is, of course, due to change in speed, also a change in
kinetic energy; but, in general, this is so small in comparison with
the total loss of head that it may be neglected.
Substituting, in equation (61), for dv its value as obtained
from the equation
pv=RT,
we find
RT s2
^p—^m^'^ (62)
Under steady flow the mass of air passing any section, for a
given interval of time, is a constant throughout the entire length
of pipe. Hence, we have, for the speed of flow,
-f-f/: m
where M is the mass passing any section per unit time, v the volume
per unit mass, and A the cross-sectional area of the pipe. Sub-
stituting the value of «, as given by equation (63), in equation
(62), we find
from which
X^PdP—f2^KA^j^ dL; .... (64)
where pi and p2 are, respectively, the initial and final pressures,
and L the length of the pipe. Finally, integrating, as indicated
in equation (64), we find
From equation (63) we have
Pi*=— 2.42-; (66)
COMPRESSED AIR AND COMPRESSORS 257
wbere «i is the initial speed. Dividing equation (65) by equation
(66), member by member, we find
pi2 '^gKRT ^^^^
Solving equation (67), respectively, for p2, «i, and/, we find
^M'-^y <^>
«,=(£i
Pi
2 ^ fj. f > . • . • (69)
and
By means of equations (65), (66), (68), and (69), the necessary
calculations, for any given case, may be made; and, by means of
equation (70), the coefficient / may be foimd for a given set of
observations.
The ratio A/P is, for cylindrical pipes, a function of the diam-
eter only; i.e..
We may substitute, then, in equation (68), the following con-
stants: g=32.2, X=Z)/4, and iJ = 53.3, and find
P2=Pi/l ^ nrV^Pi(^-7^k^V' (71)
.3x-r) ^ ^^^^^^ '
32.2 X 53.3 X -7
4
In a similar manner, the various equations may be simplified.
Equation (71) is, perhaps, best illustrated by assiuning a
concrete case, and solving for the terminal pressure. As an
example, let it be required to find the final pressiu'e, for the case
when the initial pressure is six atmospheres, the temperature
258 THERMODYNAMICS
62*^F., the quantity of air required 1200 cu.ft. per minute, the
length of pipe 5 miles, and the diameter of the pipe is 1 ft.
First of all, from equation (71), it is obvious that the pressure
may be specified in any units whatsoever.
From a series of observations made by Riedler and Gutter-
muth upon the compressed air system of Paris, extending over a
distance of about 10 miles, the diameter of the cast-iron pipe
being very nearly 1 ft. (exactly 300 mm.), Professor Unwin
deduced for the coefficient /, in this particular case, the value of
0.0029.* It must be remembered that this is not the coefficient
for a straight piece of cast-iron piping; including as it does,
bends and joints, and also a small amount of leakage. Though
transmissions to such distances are unusual, the value just quoted
for the coefficient is probably a good average value to use for a
practical case for the same diameter of piping. That is, in
any practical case, for piping of an equal diameter, we should
probably find the average losses per given length, approximately
the same.
From the conditions we have pi=88.2 lbs. per square inch,
T=522, L=26,400ft., D = l ft., and si = (1200/60)/ =25.5
4
ft. per second. For /, we will use the value 0.0029. Substituting
these values, in equation (71), we find
cQo/i 0.0029X25.5 X26,400 \i .^ ^ „
P2 = S8.2\1 429X522 ) ^^ ^' ^^'
Thus giving a loss in pressure of about 11.8 per cent. It must,
however, not be understood from this, that the percentage loss of
power in transmission is also 11.8 per cent. The efficiency of
transmission is found by taking the ratio of the work which the
air motor can do in expanding adiabatically from the pressure
P2 to that of the atmosphere, to that which would have been
obtained had adiabatic expansion taken place before transmission.
* "On the Development and Transmission of Power," by W. C. Unwin.
COMPRESSED AIR AND COMPRESSORS 259
That is, the efficiency of transmission is
_ n-V \ \p2/ J _ \P2/ _ ,^2)
Substituting in equation (72), for pa, pi, and p2, respectively,
14.7, 88.2, and 77.8, we find, for the efficiency of transmission.
i'(—Y
\77.8/ ^ ^ . .
1? = ^=0.945;
1
/14.7\7
V88.2/
where n is assumed equal to 1.4.
It is thus seen that, though the loss in pressure is about
11.8 per cent, the loss in power, due to this loss in pressure, is
only about 5.5 per cent.
The efficiency of transmission may also be defined, depending
upon the point of view, as the ratio of the work that could be
realized, before transmission, by allowing the air to expand
isothermally, to that which would be realized by means of isother-
mal expansion after transmission. In any case, for pressures such
as are ordinarily employed, the value found, for the efficiency of
transmission, by this comparison will not differ materially from
that found by means of equation (72). If we make the computa-
tion for this particular case, we find, by assuming isothermal
processes,
, 77.8
^""^li^
^'=— 88:2 = 0-«30;
which differs approximately, only 1.5 per cent from the value
found by comparing adiabatic processes.
260
THERMODYNAMICS
In the case of water, the coefficient /, other things being equal,
is a constant for all diameters. This, however, is not the case
for gases. In the case of air, the coefficient / is some function of
the diameter. Various empirical formulsB have been proposed,
for cast-iron piping, by means of which / is f oimd, in terms of the
diameter. None of them, however, are true for all diameters.
As an example, the following formula, proposed by Professor
XJnwin, may be cited. According to this formula, the coefficient
is
/=0.0027(l+j|g).
However, by computing the coefficient, for various diameters,
by means of this formula, and comparing with the values, as found
by actual experiments, it is foimd that there is considerable
discrepancy, as the following table will show:
D.
In Feet.
Coefficient.
By Experiment.
By Formxila.
0.492
0.656
0.980
0.00449
0.00377
0.0029
0.00435
0.00393
0.00351
For the two smaller diameters there is very close agreement;
but, in the case of the one of 0.98 ft. diameter, the discrepancy is
considerable. Professor Unwin has proposed the value 0.003 for
all diameters of 1 ft. or over.
184. Composite Diagram. We are now prepared to show,
by means of the p-v diagram, the losses for the compressor, the
line, and the motor. Let, in Fig. 50, 01 and OH represent, respect-
ively, the axes of zero pressure and zero voliune; and the line
AB the aspirating stroke. Assume further that the compressor
is one working on two stages, compressing first adiabatically,
in the low-pressure cylinder, from the pressure pa, as represented
COMPRESSED AIR AND COMPRESSORS
261
by the point S, to a pressure p, as represented by the point C.
At the point C, the exhaust-valve of the low-pressure cylinder
opens and the air is expelled to the receiver at the constant pres-
sure p.. In the receiver, the temperature falls to its initial value,
and \he volume shrinks by an amoimt represented by CD; the
point D being on the isotherm BF. The condition of the same
mass of air now, as regards pressm-e and volimae, at the end of
the aspirating stroke, in the high-pressure cylinder, is represented
by the point D. Compression now takes place adiabatically
from the point D, to the point S, to a pressure pi. When the pres-
H
sure pi has been attained, the exhaust-valve opens and expulsion
takes place under constant pressure, as represented by the line
EG. In the reservoir, the temperature of the air falls to its initial
value, and the volume shrinks by an amount EF; the point F
being on the isotherm BDF. During isothermal transmission,
the pressure falls by an amoimt represented by GK; and at the
end of admission in the air motor, i.e., at the point of cut-off, the
condition of the air, as regards pressure and volume, is represented
by the point L. The point L is again on the isotherm BDF,
From the point of cut-off, L, the air expands adiabatically, as
represented by the curve LM. At the point M release occurs;
and the expulsion stroke is represented by the line MA»
From an inspection of the figure it is evident that the net
work done, per cycle, by the compressor, is measured by the area
262 THERMODYNAMICS
ABC DEO; and, the net work recovered by the air motor, per
cycle, is measured by the area KLMA. Hence, the total loss of
work is measured by the area BCDEOKLM. The work lost,
per cycle, due to the compression and expansion not being isother-
mal, i.e., in the compressor and motor, is measured by the area
BCDEFLM; and, the work lost in transmission is measured by
the area FGKL. In general, the thermodynamic loss in the com-
pressor and motor is large, in comparison with the loss, due to
friction, during transmission.
186. Theoretical Efficiency of System. It will now prove
instructive to assume a concrete case and make, without consider-
ing other losses, a comparison between the three losses; that
is, the thermodynamic loss, due to the compression, in the com-
pressor, being adiabatic in place of isothermal, the loss in trans-
mission, and the thermodynamic loss, due to the expansion in the
motor, being adiabatic in place of isothermal.
Let it be required to compress the air to a pressure of six atmos-
pheres by means of a two-stage compressor. The work required,
per pound of air, according to equation (23), will be
n-l
-"^Mifr-^]
=^X27,800(6^-1) = 56,800 ft.-lbs. per pound.
If the expansion now take place isothermally, after cooling, the
work recovered will be
W2 = PaVa log ^- = 27,800 log 6 = 49,800 ft.-lbs. per pound.
Pa
This is a loss of about 12.3 per cent.
Assimie, now, that the transmission line has the same constants
as that discussed in Art. 183. Then the pressure at the end of
COMPRESSED AIR AND COMPRESSORS 263
the line will be 77.8 lbs. per square inch; and the work that can
now be recovered, due to isothermal expansion, will be
1^8 = 27,800 log j^= 46,300 ft.-lbs. per pound.
This is a loss of about 6.2 per cent of the total work.
If the expansion now takes place adiabatically, the work done
on the air motor is
Tr3 = 27,800xJ^{ l-(^y } =36,900 ft.-lbs. per pound.
This gives a loss in the air motor of about 16.6 per cent of the total
work done.
We have then, the following : Per Cent.
Loss in compressor 12.3
Loss in transmission 6.2
Loss in motor 16.6
Efficiency of system. . . . , 64.9
Total 100
From the foregoing computations, it is obvious that the
efficiency of the system is low, not due to the loss in transmission;
but on account of the combined losses in the compressor and
motor.
Assume, now, that the air is reheated to a temperature 275 °F.
above the smroundings. The work which the air will now do
on the motor is
T 7Q7
Tr4=—Tr3=i|ix 36,900 =56,300 ft.-lbs. per pound;
where T is the temperature to which the air is heated before
bdng admitted to the motor. This gives, for the gain in work,
for the same quantity of air consumed, by the motor, approxi-
mately 52.6 per cent.
To make a comparison now, between the work done on the
motor and that done on the compressor, it will be necessary to
264 THERMODYNAMICS
add to the work done on the compressor, the work due to the heat
consumed in reheating the air. The heat consumed in elevating
the temperature of 1 pound of air from the temperature Ti to
the temperature T, is
Q=Cp(r-ri)B.T.U.
And the work which would be realized on a Camot cycle, is
Substituting the various values, we find
Tr6=778X0.238X275X^= 17,600 ft.-lbs.
Taking the ratio now, of Wa to the sum of Wi and W5, we find,
for the efficiency of the system, 75.7 per cent, as against 64.9 i)er
cent, obtained without reheating. Commercially, however, the
gain is greater than that indicated by the computations. For,
as previously stated, a low-grade fuel may be employed, and the
motor operates better, especially so if a small percentage of water
is injected into the heater. This water is evaporated in going
through the heater, and condensed in going through the motor.
There is involved m this operation a small thermodynamic loss;
but otherwise, the effect is good, since the water present helps to
prevent leakage. Finally, reheating has the effect of increasing
the capacity of both the compressor and line.
It must be emphasized that in no case are efficiencies obtained
as high as those indicated by the foregoing computations. Due
to imperfect valve action, leakage, and mechanical losses, in both
the compressor and motor, the efficiency of the system may be
reduced by 10 to 15 per cent below that indicated by the com-
putations.*
*For actual tests on air transmission BystemBy see Unwin, "On the
Development and Transmission of Power."
CHAPTER XVII
REFRIGERATION
186. The object of refrigeration is to maintain the temperature
of some body; or aggregation of bodies, at some point lower
than that of the smroundings. This may be done in two ways.
One method is to abstract heat directly, by means of a refriger-
ating machine, from the medium surrounding the bodies. The
other method is to bring about the desired lowering of temperature
by means of ice. The ice employed, to bring about the desired
refrigeration, may be harvested, during the cold season, from
rivers and lakes, or else, the so-called " artificial ice," produced
by means of refrigerating machines, may be used.
Since the putrefaction of organic growths, such as foodstuffs,
is retarded with lowering of temperature, and, in general com-
pletely prevented when the temperature becomes sufficiently
low, the prime object of refrigeration is not the maintaining of
low temperatiu'es, but rather the effects due to such low temper-
atures; i.e., the preservation of foodstuffs during storage and
shipment, and, in general, the promotion of health and comfort.
187. Commercial Refrigerating Machines. Refrigeration may
be brought about in various ways. But commercially successful
refrigerating machines are restricted to two types; viz, refriger-
ating machines in which air is the working substance, and machines
in which some volatile liquid, such as anunonia, or carbon-dioxide,
is employed as a working substance. For ammonia machines,
there are again two distinct methods of operation; viz, com-
pressor machines, and absorption machines. These various types
will be discussed subsequently under separate headings.
265
266 THERMODYNAMICS
All commercial refrigerating machines operate as reversed
engines; but, it must not be understood from this that the machine
is reversible. The working substance abstracts heat from a
body of, relatively, low temperature, called the refrigeratory con-
sumes energy either in the form of mechanical work or heat,
and rejects heat to a condenser or cooler. The heat rejected to
the cooler, barring various losses, is equal to the heat taken from
the refrigerator plus the heat equivalent of the energy consumed
in bringing about the transfer.
Equation (24), of Chapter VIII, states that, for an engine
operating reversed, on a Camot cycle,
W^JH2^; (1)
where W is the energy consumed in bringing about the transfer,
H2 the heat abstracted from the refrigerator, S the temperature
of the source, and R the temperature of the refrigerator. The
source, in the case of a reversible engine, corresponds to the cooler
of a refrigerating machine.
In discussing the Camot cycle, it was foimd that, other things
being equal, the greater the range in temperature, the greater the
amount of work realized for a given quantity of heat abstracted
from the source. On the other hand, equation (1) clearly indi-
cates that, other things being equal, for a given quantity of heat
H2i abstracted ftom the refrigerator, the wo k which must be
done by the compressor decreases as the difference of temperature
between the cooler and refrigerator is decreased. Hence, the range
in temperature between refrigerator and cooler, for refrigerating
machines, should be as small as possible.
188. Air Refrigerating Machine. The air refrigerating system
consists essentially of four parts; viz, a cold storage room, a
compression cylinder, an expansion cylinder, and a cooler. The
cycle is as follows : During the aspirating stroke, of the compressor
piston, air flows into the cylinder, from the cold storage room,
REFRIGERATION 267
which during the return stroke is compressed, practically adia-
batically, to the desired pressure, and expelled to the cooler.
The cooler, usually, consists of a series of pipes in which the air is
cooled by water circulating through the tank in which the pipes
are placed. From the cooler the air passes into the expansion
cylinder, where it does work on the piston, expanding practically
adiabatically, and is finally exhausted, at a low temperature, to
the cold storage room. The work done in the expansion cylinder
is utilized in helping to drive the compressor. Hence the work,
barring mechanical losses, which must be supplied to the com-
pressor by means of some motor, is the difference between that
done in the compression cylinder and that done in the expansion
cylinder.
189. Ideal Coefficient of Performance. To make a mathe-
matical discussion, of the cycle just described, it will be necessary
to assume ideal conditions. Let T2 be the temperature of the
air entering the cooler, at the end of the adiabatic compression,
Ta its temperature as it leaves the cooler and is admitted to the
expansion cylinder, To its temperature at the end of the adiabatic
expansion as it enters the cold storage room, and Ti its temper-
ature as it leaves the cold storage room and enters the compressor.
It will now be assumed that the pressures in both the cooling pipes
and cold storage room are constant throughout the cycle, and the
machine is mechanically perfect.
By equation (15), Art. 177, we have for the work done per
pound of air, on the piston of the compressor,
Wi=JCp(T2-Ti) (2)
By equation (48), Art. 181, we have for the work done per pound
of air, on the piston, in the expansion cylinder,
W2 = JCp{Ta-To) (3)
268 THERMODYNAMICS
The work which must be supplied, to make the process take place,
is the difference between Wi and W2; i.e.,
Wz=JCp{{T2'-Ti)'-{Ta-To)] (4)
Since, according to the assumed conditions, the ratio of the ranges
in pressures for the two cylinders are the same, we find
r.4: («
Substituting the value of To, as given by equation (5) in equation
(4) we find
Wz=JCAT2-Ti)^^^ (6)
The heat per pound of air, expressed in mechanical units, taken
from the refrigerator, is
W^^JCpiTi-To).
Substituting again, for To, its value, we find
Tr4=JC,^^(r2-ra) (7)
Taking the ratio of W^ to Wz we find, for the ideal coefficient of
performance,
^ Wz T2-T1 ^^^
Equation (8) again shows that, the smaller the difference in
temperature between refrigerator and cooler, the larger will
become the ratio of the work equivalent of the heat abstracted
from the refrigerator, to the work supplied. It is, of course,
obvious that, due to the fact that it is practically uneconomical
to construct cooling pipes of sufficient volume, such that the pres-
sure throughout the cycle in the cooler is constant, and further,
since the pressure in the cold storage room varies, the ratio,
REFRIGERATION 269
as expressed by equation (8), cannot be realized in practice.
Furthermore, due to various losses, which must be experienced,
in the case of an actual refrigerating machine, this ratio is still
further reduced. Solving equation (8), for the work that must
be supplied to a perfect machme, we find
Ws^^wJ^^^ (9)
The commercial efficiency of a refrigerating machine may be
defined as the ratio of the work which would have to be done,
for the given range of temperature and given quantity of heat
removed from the refrigerator, on a perfect machine, to that
actually required. If Ws is the work actually required, then the
conmiercial efficiency is
Ws'Ws^ Ti ^^"^
The heat which must be carried away by the circulating water,
in the cooler, per pound of air, is
Hl^C^{T2-Ta) (11)
The cycle of an air refrigerating machine may be conveniently
represented by means of the T-? diagram. By equation (12),
Art. 169, the change in entropy, when both the pressure and volume
vary, is
92-9i = C.log2?+Cplog||? (12)
In the cycle just discussed it was assumed that the pressure,
during the absorption and rejection of heat, remains constant.
Hence, equation (12) becomes
92-(pi = Cplogg=Cplog^^; .... (13)
and the heating and cooling curves, on the T- 9 diagram, are loga-
rithmic. Let, in Fig. 51, the point A represent the condition of
270
THERMODYNAMICS
the air, as regards temperature and entropy, at the instant when
it enters the compressor at the temperature Ti. During the
adiabatic compression the entropy remains constant and the tem-
perature changes from Ti to 72, as represented by the line AB.
The cooling then takes place, as represented by the curve BC,
at constant pressure, to the temperature Ta- The adiabatic
expansion, from the temperature Ta to the temperature Tq, is
represented by the constant entropy line CD. Finally, the rise
in temperature, in the refrigerator, at constant pressure, from To
to Ti, is represented by the curve DA ; and the cycle is completed.
The heat abstracted from the refrigerator is measured by the area
FADE, the heat rejected to the cooler is measured by the area
FBCEj and the work done, on the compressor, is measured by the
area A BCD. Finally, the ideal coefficient of performance is
given by
Aresi FADE
V =
Aresi ABCD'
Since the rejection of heat to the cooler, and the abstraction
of heat from the cold storage room, both take place at constant
REFRIGERATION 271
pressure, equation (8) may be deduced in a very simple mann^.
If the heat abstracted from the cold storage room, for a given
interval of time, is
ff2=Cp(ri-ro),
then the heat rejected to the cooler for the same interval of time, is
Hi = Cp(T2 — Ta) •
Therefore, the ideal coefficient of performance is
H2 C pCri-fo) ^ Ti .
"^ H1-H2 C,{T2-Ta)-Cp{Ti-To) T2-T1'
which is the same as previously foimd.
It must be emphasized that the equations deduced, in this
article, do not represent conditions as found in actual practice.
For the pressure, in the cooling pipes, of any actual refrigerating
machine will vary considerably throughout the cycle. Hence,
the actual coefficient of performance, even when all other losses
are neglected, will be less than that indicated by equation (8).
Due to the fact that air has a low thermal capacity, air refrig-
erating machines are necessarily bulky, and therefore, commer-
cially imeconomical. However, there are certain places, as for
example on board of ships, where it is inadvisable to use machines
employing a volatile liquid. For, in the first place, there are possi-
ble dangers from injurious escaping gases. But, even if the escap-
ing gas is not injurious, there is always the possibility of a large
leak, and consequently a total loss of the working substance, which
cannot be replaced imtil the end of the trip. This, however,
means a complete disablement of the plant. Hence, air machines
are used only as a matter of expedience and not economy, in place
of refrigerating machines employing a volatile liquid as a working
substance.
190. Compression Machines Using Volatile Liquids. Com-
pression refrigerating machines, using a volatile liquid for the
272
THERMODYNAMICS
working substance, consist essentially of the parts as represented
diagrammatically, in Fig. 52. A is the compression cylinder
where the vapor is compressed, and then expelled into coils
immersed in water in S; B being the condenser, or cooler. If
the vapor is just saturated as it leaves the refrigerating coils,
superheating may take place, during compression; this however
is usually very small in comparison with the heat of condensation.
Due to the high pressure in the condenser, and the low temper-
ature, maintained by the circulating water, the vapor condenses,
gives up the superheat and heat of condensation, which is carried
-J
Wl^J VJJIV/
rvMfVVyA, s*AAV» \ia^
\y «AX^1A«V* M^-t^ *T tj Aa^VXJ
D
4
Va^ V tJ VV^A »«^^^ VI
A
p
B
B
i:
G
^
D
Fig. 52.
In the tank C, the liquid is under a pressure corresponding to that
of its vapor, for the existing temperature; the temperature of
the liquid in the storage tank, usually does not differ materially
from that of the surroundings. As an example, if the liquid
employed be ammonia and the temperature in the tank is 76**F.,
then the pressure of the vapor is approximately 140 lbs. per square
inch. Due to this high pressure, under which the liquid is in C,
it flows, through the expansion valve D, into the coils in the refrig-
erator E, The pressure in the coils, due to the aspirating action of
the compressor, is low. By regulating the expansion valve, or
the speed of the compressor, or both, the pressure in the refriger-
ator coils may be varied at pleasm*e. Since, when the liquid passes
through the expansion valve, the process is adiabatic, and no work
REFRIGERATION 273
is being done, the total heat content remains the same. There-
fore, for thermal equilibrium to obtain, when the pressure falls
from pi, that existing in the storage tank, to p2, that existing in
the refrigerating coils, there must take place a certain amount
of. evaporation, such that
hi=h2+qr2; (14)
where hi and A2, respectively, are the heats of the liquid corre-
sponding to the pressures pi and p2, q the amount of dryness, and
r2 the heat of vaporization at the pressure p2. From equation
(14), we find
«=^; (15)
and the remainder of the liquid can, then, if completely vapor-
ized, take from the surrounding medium the quantity of heat
ff2 = (l-(?)r2 (16)
In order that heat may flow from the medium in E, into the
coils it is, of course, necessary that the temperature of the mediimi
be higher than that of the liquid, in the coils, corresponding to the
pressiu-e p2. If the difference of temperature is suflScient, the liquid
will be completely vaporized; and the quantity of heat, as
expressed by equation (16), will be removed from the refrigerator.
If the difiference of temperature be greater than this, the vapor
becomes superheated; and the quantity of heat removed from the
refrigerator will be greater than that indicated by equation (16).
The ideal p-v diagram, of the cycle just discussed, is represented
in Fig. 53. The point A represents the condition, as regards
pressure and volmne, of the vapor at the beginning of the aspira-
ting stroke, and the point B represents the condition at the end
of the aspirating stroke; the line AB, therefore, represents the
volume, due to complete vaporization under constant pressure.
The curve BC represents the compression, which is nearly adia-
274
THERMODYNAMICS
batic, CD represents the expulsion, and also the condensation,
under constant pressure, in the condenser, and DE the change in
pressure, and consequent change in voliune, due to partial
evaporation in passing through the expansion-valve. Therefore,
the net work done, during the cycle, is measured by the area
A BCD, Fina ly the ideal coefficient of performance is given by
the ratio of the work equivalent of the heat removed from the
refrigerator to the work equivalent of the area A BCD.
The refrigerating coils, in which the vaporization takes place,
may be placed directly in a cold storage room, in the form of pipes,
or else placed in a tank containing a solution of some salt, called
brine. The freezing point for the brine must, of course, be lower
than the temperature in the coils. The brine may then be em-
ployed, by circulating it through pipes, to bring about refrigeration
in some place remote from the plant, or else, to produce ice, by
abstracting heat from water, contained in tanks, immersed in
the brine.
The cycle of a compressor refrigerating plant, using a volatile
liquid £us a working substance, is most instructively represented
by the T— 9 diagram. However, before plotting the T— 9 dia-
gram, it will be necessary to deduce an expression for the change
REFRIGERATION 275
in entropy, for the substance, when passing through the expan-
sion valve. To do this, let Ti be the temperature of the liquid
in the storage tank, then by assuming some arbitrary temperature,
say To, from which the entropy is measiu'ed, the entropy of a
unit mass of the Uquid, before passing through the expansion
valve, is
9i = cl -77=cIogjr; (17)
where c is the thermal capacity, of the liquid, per unit mass.
Assume some temperatiu*e T, in the refrigerating coils; T being,
of course, less than Ti. The entropy, then of a unit mass of
liquid and vapor, measured from the same zero, is
92
= cj y+^=clog jr+^; .... (18)
where q is the amount of dryness, and r the heat of vaporization
corresponding to the temperature T, Taking the difference
between equations (18) and (17), we find, for the change in entropy
in passing through the expansion valve,
9=92- 9i = c log ^-c log j^+^. . . . (19)
But,
gr=Ai-/i=c(Ti-r);
substituting this value of gr in equation (19), and simplifying,
we find
9=c(log|^^+y'-l) (20)
Dififerentiating equation (20) with respect to Tj Ti being assumed
constant, we find
%--S-') <^'>
Equation (21) shows, since T\>T, that as the temperature
increases, the entropy decreases, and vice versa. Hence, the
276
THERMODYNAMICS
entropy of the substance is increased by passing through the
expansion valve. This is necessarily so, since the process is irre-
versible.
The T- 9 diagram, Fig. 54, indicates the various parts of the
cycle. BC is the constant entropy line for the adiabatic com-
pression of the vapor, from the temperature T2 to T,; if the vapor
be just saturated, as shown, when the compression begins, it will
become superheated during compression. CK represents the
cooling of the vapor to the temperature of condensation, Ti;
7
C
D
Ti
»/
A
T,
1
\
\
\
*
B
F! Gj
H
9
Fig. 54.
and KD represents the condensation of the vapor, in the con-
denser, at the constant temperature Ti. DA represents the tem-
perature entropy curve for the cooling of the liquid, without
expansion, from the temperature T\ to ^2. Had evaporation
taken place, without expansion, after cooling along the curve DA,
the quantity of heat removed from the refrigerator would be meas-
ured by the area FABH. Due to expansion, however, through
the expansion valve, the entropy of the substance increases in
changing from the temperature Ti to ^2, as indicated by the curve
DE. The curve DE is determined by assiuning various values
of temperature, between Ti and T2, and solving, by means of equa-
REFRIGERATION 277
tion (20), for the corresponding entropy. Hence the heat that it
is now possible to remove from the refrigerator, in bringing about
complete vaporization, is measured by the area 6EBH. Con-
sequently, the amount of refrigeration that is lost, due to the
change in entropy, in passing through the expansion valve, is
measured by the area FAEG. Had there been superheating in
the refrigerating coils, the quantity of heat removed from the
refrigerator would be increased; but, due to this superheating,
the vapor at the end of the compression will, likewise, be super-
heated by an additional amount.
Since the evaporation, which takes place while the liquid
passes through the expansion valve, has no refrigerating value,
but merely brings about thermal equilibrium, by reducing the
temperatiu-e of the liquid to that existing in the coil, it follows
that the change in entropy, along the curve DE^ depends upon
the ratio of the heat of vaporization to the thermal capacity of
the liquid. The higher the ratio of the heat of vaporization to
thermal capacity of liquid, the smaUer the amount of vapomation
required, for a given difference of temperatiu-e, to reduce the tem-
peratiu*e of the liquid to that existing in the coil ; and conseiquently
the smaller will be the area FAEG. Therefore, since the work
done by the compressor is independent of the amoimt of vapor-
ization that takes place, along the ciure DE, it follows that a
liquid for which the ratio, of heat of vaporization to thermal
capacity, is high, is best suited, from an economical standpoint,
for refrigerating purposes.
191. Absorption Machines. The principle of operation of an
absorption refrigerating machine is based on the fact that the volume
of ammonia vapor that can be absorbed by a given volume of
water, other things being equal, depends upon the temperature,
and decreases rapidly as the temperature is increased. Hence
if water, at a low temperature, is saturated with ammonia vapor,
then, to drive off the vapor, heat must be absorbed by the water.
Likewise, if ammonia vapor be passed into water at a low tern-
278
THERMODYNAMICS
perature, absorption will take place with a consequent develop-
ment of heat.
An absorption refrigerating machine is represented, diagram-
matically, in Fig. 55. A is a storage tank containing anmionia
from which expansion takes place through the valve B, into
refrigerating coils in C, where refrigeration takes place. From
the coils in C, the ammonia vapor passes into the liquid in the
absorber D. The liquid in D is a solution of ammonia in water,
of slight concentration and, relatively, low temperature. ThQ
H
^^9
:E =
' Bp
FiQ. 55.
liquid in the absorber being at a low temperature and only slightly
concentrated, the incoming vapor is readily absorbed. The liquid
of high concentration is removed from the bottom of D, by means of
the pump P, and forced, at a, into the generator F. In the gener-
ator is placed a heating coil H, by means of which heat is supplied
to the highly concentrated solution, and raises its temperature.
Due to the high temperature, part of the vapor is expelled from the
solution, under a high pressure, and passes into the condenser O.
The condenser is maintained at a, relatively, low temperature by
means of circulating water. Due to this low temperature and the
high pressure, the vapor condenses and flows into the storage
tank A. By means of the valve /, the pressure in G and A is regu-
REFRIGERATION
279
lated. Finally, the solution of low concentration, at the bottom
of the generator F, is forced, due to the high pressure subsisting
in the generator, into the absorber, at b. The pipes which convey
the highly concentrated solution into the generator at a, and the
solution of low concentration into the absorber at 6, both pass
through the heat exchanger E. In the heat exchanger the solution
at a low temperature, going from the absorber to the generator,
takes up heat from the high temperature solution, going from the
generator to the absorber.
The cycle is, then, as follows : The absorption, in the absorber,
corresponds to the aspirating stroke of the compressor, as repre-
FiG. 66.
sented by AB of Fig. 66. The change in pressure in going from
the absorber through the generator is represented by the curve
BCf and corresponds to the compression curve of the compressor.
The line CD represents the condensation at constant pressure,
the same as in Fig. 53. Finally, the curve DE represents the fall
in pressure and consequent increment in volume, due to partial
evaporation of the liquid, in passing through the expansion-
valve B. There is, of course, in this cycle, as well as in the com-
pressor cycle, due to a partial evaporation when the liquid passes
through the expansion-valve, a loss in refrigeration. Furthermore,
there is, due to the fact that the ammonia vapor when distilled
in the generator, carries with it a certain amount of aqueous
vapor, an unavoidable loss.
280
THERMODYNAMICS
Thermodynamically speaking, the ideal coefficient of per-
formance of the absorption machine, as well as that of any other
refrigerating machme, is given by
R
where S is the temperature of the condenser, and R the tempera-
ture in the refrigerator. On the other hand, the commercial
efficiency is given by the ratio of the work which would have to
be done on a perfect engine, to bring about the given transfer of
heat, to the energy actually consmned. That is, if ^2 is the quan-
tity of heat abstracted from the refrigerator, during a given
interval of time. Hi the quantity of heat supplied to the generator,
and W the work done on the pump, during the same interval
of time, the commercial efficiency is given by
JHi
&=
S-R
R
JHi+W
(22)
The following table, taken from " Landolt and Bdmstein,"
is given to show how the coefficient of absorption for ammonia
vapor, under normal pressure, varies with the temperature:
T
K
T
K
98.7
15
60.6
1
92.7
16
59.1
2
87.7
17
57.6
3
83.6
18
56.1
4
79.9
19
54.7
6 •
77.3
20
53.5
6
75.6
21
51.9
7
73.9
22
50.6
8
72.3
23
49.6
9
70.6
24
48.6
10
68.9
25
47.6
11
67.2
26
46.5
12
65.5
27
45.5
13
63.7
28
44.4
14
62.1
29
43.4
REFRIGERATION
281
where t is the temperature in degrees centigrade, and K is the
number of grams of ammonia vapor absorbed per 100 c.c. of
water.
By heai of dilviion of a substance is meant the quantity of
heat which is evolved when a unit mass of the substance is diluted
to an extent such that practically no more heat is evolved upon
further d ution.
According to experiments by Berthelot, when 1 gram of
liquid anunonia has been dissolved in n grams of water, and this
solution is then fully diluted, the heat evolved is as given in the
following table:
n
Gram Calories.
n
Gram Calories.
1.04
1.06
1.13
1.98
75.6
74.4
68.8
40.0
3.18
3.76
6.11
10.1
22.6
18.8
12.3
0.12
The results given in the table were obtained from experiments
conducted at temperatures of 14°C.
By inspection it is seen that very little heat is evolved after
the dilution is greater than 10 to 1. It has been proposed to
employ the empirical equation
n
(23)
for the heat of dilution when a solution, of 1 gram of ammonia
dissolved in n grams of water, is fully diluted; H being the heat
evolved, and h some constant. If equation (23) be applied to
the values as given in the table, the value 78 be assigned to A,
and the values of H computed and compared with the observed
values, an idea will be obtained as to how closely the empirical
equation conforms to the actual experimental results.
282
THERMODYNAMICS
n
Observed.
Computed.
n
Observed.
Computed.
1.04
1.06
1.13
1.98
75.6
74.4
68.8
40.0
75.0
73.6
69.0
39.4
1 3.18
3.76
6.11
10.1
22.6
18.8
12.3
0.12
24.5
20.7
12.8
7.7
The foregoing table shows that, when the mitial dilution is
not greater than 6 to 1,. equation (23) gives fairly consistent
results; but, for initial dilutions greater than this, the equation
breaks down completely. Furthermore, since the lowest initial
dilution in Berthelot's experiments was 1.04 to 1, equation (23)
is necessarily doubtful for initial dilutions lower than this.
Experiment shows that, if 1 gram of ammonia vapor is
absorbed by water and completely diluted, there is evolved a quan-
tity of heat equal to 496 gram calories; hence, if m grams of
ammonia are absorbed, and complete dilution take place, there
will be evolved 496m gram calories. Therefore, if we assume
equation (23) to hold, there will be evolved, when m grams of
ammonia vapor are absorbed by n grams of water,
Hi^mQ-^ = m(Q--h); .... (24)
n/m \ n / ^
where n/m is the number of grams of water per gram of ammonia,
and Q the quantity of heat evolved when 1 gram of ammonia
vapor is absorbed by water and completely diluted. If, now,
m+k grams of ammonia be absorbed by n grams of water, the
number of grams of water per gram of ammonia will be n/{m+k).
Therefore, the quantity of heat
ff2 = (m+A;)(Q-^XA), . . . .
(25)
will be evolved. Taking the difference between the right-hand
members of equations (25) and (24), for the quantity of heat
REFRIGERATION 283
evolved, when a solution containing m grams of ammonia to n
grams of water, absorbs k grams of ammonia, we find
=A;jQ-^(2m+fc)L . . (26)
Substituting for Q and h the numerical values, we obtain
H^k
78
496-— (2m+fc)
gram calories. . . (27)
Equation (27) may be reduced to English units as follows:
£r=A;]893-^(2m+fc)
B.T.U (28)
That is, equation "^r^ is the expression for the heat, in B.T.U. ,
which is evolved when a solution containing m pounds of ammonia
to n poimds of water, absorbs k pounds of ammonia vapor.
As previously stated, the foregoing equations are empirical
and are true only between certain limits of initial dilution; further-
more, since the heat of absorption and dilution varies with the
temperature, the results obtained by means of these equations
are to some extent doubtful. The equations have been deduced
merely to show the method of attack.
Under ideal conditions the heat developed in the absorber is
equal to that required in the generator; but, since the temperature
of the generator must be higher than that of the absorber, it is
impossible to utilize the heat developed in the absorber. Hence,
to maintain the process, heat must be suppHed, by means of some
independent source, to the high-temperature generator, and heat
must be abstracted from the low-temperature absorber.
In the case of a compression machine the energy consumed
varies directly as the difference of temperature between the con-
denser and refrigerator. This, however, is not so in the case of
Cri^tm
284 THERMODYNAMICS
an absorption machine; hence, for a wide range in temperature,
the absorption machine is thermodynamically superior. A
further advantage, which is mechanical, is that no compressor
is required. In certain cases the heating in the generator is brought
about by means of exhaust steam, from engines, which is again
economical. Finally, the power consumed by the pump in an
absorption machine is small in comparison with the other quan-
tities involved.*
192. Comparison of Air and Axnmonia Machines. It was
stated in Art. 189 that, due to the low thermal capacity of air,
refrigerating machines employing air as a working substance are
necessarily bulky. It is impossible to make a general comparison;
but a rough estimate may be obtained by assumi!ng a concrete
case. Let it be assumed that the temperature of the refrigerator
is 32°F., and that the range in temperature of the air in passing
through the refrigerator is lOO^F. One pound of air will then
remove, from the refrigerator,
CpCri-To) =0.238X100=23.8 B.T.U.;
and to do this, the compressor must take in 12.4 cu.ft. The volume,
per B.T.U. removed from the refrigerator, then is
12.4/23.8=0.521 cu.ft.
Assume, now, an ammonia compression machine, with a temper-
ature of 70°F. for the reservoir. The dryness after passing
through the expansion valve will be
^ r2 540 '
and the quantity of heat that can be removed, by complete
vaporization taking place at 32°F., is
(l-g)r2 = 540(l-0.078)=498 B.T.U. per pound.
*Fora comprehensive discussion of absorption machines see ''Modern
Refrigerating Machinery'' by Hans Lorenz.
f From Peabody's Steam Tables.
REFRIGERATION 285
The specific volume of ammonia vapor, at 32®F., is approximately
4.74 cu.ft. per pound. Hence we find, for the volume, per B.T.U.,
4.74/498=0.00952.
Taking the ratio of the volimie for air, to that for ammonia, we
find
0.521/0.00952=54.7.
This shows that for the assumed conditions, other things being
equal, the bulk of the compression cylinder of an air refrigerating
machine is very large in comparison with that of an ammonia
machine; but, further than this, the air machine must also have
an expansion cylinder. For lower temperatures in the refriger-
ator, the ratio of the two volumes becomes somewhat less. Assume
the temperature of the refrigerator 15°F., then the dryness, after
passing through the expansion-valve, is
61/554 = 0.110;
and the quantity of heat that can be removed, by complete
vaporization, is
554(1-0.110) =493 B.T.U. per pound.
The specific volume for the vapor of ammonia at 15°F. is 6.68
cu.ft. per pound; hence, we find for the volume, per B.T.U.,
6.68/493=0.0135 cu.ft.
The volume of air that the compressor must now take in, at the
temperature of 15°F., is approximately 12 cu.ft. ; hence the volmne
of airperB.T.U., is
12/23.8=0.504 cu.ft.
Taking the ratio of the volmne for air, to that for ammonia, we
find
0.504/0.0135 = 37.3.
The ammonia machine is also superior from the thermody-
namic standpoint. By considering the two cycles, it is obvious
286 THERMODYNAMICS
that the cycle of the ammonia refrigerating machine, approaches
the Carnot cycle much more closely than does the cycle of an air
machine. For, in the ammonia cycle, the greater part of the heat
is abstracted and rejected, respectively, dm-ing vaporization and
condensation; i.e., at constant temperature. On the other
hand, in the case of the air cycle, both the abstraction and rejec-
tion of heat take place with continuously varying temperature.
The foregoing may be illustrated roughly as follows : As previously
shown, the work done by a compressor per cycle, if the compres-
sion is adiabatic, is
Wi = JCp{T2 - Ti) per pound.
The heat, expressed in mechanical imits, removed by 1 pound
of ammonia, from the refrigerator, is
W2=Jr(l-q);
and the ideal coefficient of performance is
r_W2_^ r(l-g) . .
^ Wi C^{T2-Ti) ^^^^
It was shown, in Art, 189, that the ideal coefficient of per-
formance of an air refrigeration machine is
. ^" = 7^ (30)
In equations (29) and (30), Ti and T2 are, respectively, the
temperatures before and after adiabatic compression. If we
assume, now, that the ranges in temperature for the two machines
are equal, we find, for the ratio of the performance for the two
processes,
y^ e«
If we are dealing with ammonia, and conditions are such as
ordinarily obtain in refrigerating plants, then, in equation (31),
the numerator and Ti will be practically equal. But, Cp for
REFRIGERATION 287
ammonia vapor is approximately 0.53; hence, the coefficient of
performance for the ammonia machine, bb expressed by equation
(29), is approximately double that for the air machine, as ex-
pressed by equation (30).
From the foregoing discussion it is obvious that, due to its
enormous bulk, and consequent mechanical losses, together with
its thermodynamic inferiority, the air refrigerating machine is
very uneconomical, both from the standpoint of first cost and
operation.
193* The Kelvin Heating Machine. Before leaving the sub-
ject of refrigerating machines, it will be interesting to consider
a heat engine running reversed as a warming machine. This was
suggested as early as 1852 by Lord Kelvin. To illustrate this,
let it be desired to maintain the temperature of a room higher
than that of the surrounding atmosphere. This may be brought
about by the direct application of heat, or else by a heat engine
running reversed. Assume the heating to be brought about by
an air refrigerating machine, such as discussed in Art. 188, then
during the aspirating stroke a charge of air flows into the com-
pression cylinder at a temperature Ta, This charge is now com-
pressed to a temperature Ti and expelled into pipes, placed in
the room which it is desired to heat, where heat is abstracted.
After cooling, the air does work in the expansion cylinder and is
expelled to the atmosphere. For a reversible engine, the heat
rejected to the room is equal to the heat taken in from the
atmosphere plus the heat equivalent of the work done on the air.
If Hi is the heat rejected to the room, and Ha the heat taken in
from the atmosphere, then
Hi=Ha+AW:
and
W=JHi
Ti-Ta
If, now, Ti — To, the required range, be small, then the heat
equivalent of W will be a small fractional part of Hi,
288 THERMODYNAMICS
To illustrate further: Assume a situation where it is impos-
sible to obtain fuel of any kind, but that there is available energy
in the form of an electric current. Heating may then be brought
about in two ways. That is, heat maiy be developed by passing
the current through a suitable resistance, or else, the energy may
be consumed in driving an electric motor, which in turn drives
some form of reversed heat engine. In either case, the energy
spent per unit time, due to the current consumed, is given by the
product of e.m./. and current. To make a simple comparison
it will be necessary to assume certain conditions. Let the tem-
perature of the atmosphere be 0°F., and that required in the heat-
ing coils, so as to maintain the room at a proper temperature,
be 165°F. Now, to bring about equal heating effects, the heat
dissipated per unit time must be the same in each case. Let H
be the heat required per unit time, then
AEIi=H] (32)
where E is the applied e,m.f. and /i the current consumed, when
the heating is brought about by means of resistances. Assume
now, a perfect electric motor driving a perfect warming machine.
The power consumed to bring about the same heating efifects,
for the given temperatures, is
AEh^H^^^^iyi, (33)
where I2 is the current consumed by the motor. Solving by
means of equations (32) and (33), for /2, we find
showing that for a commercial eflSciency even as low as 26.4 per
cent, the warming machine is thermodjniamically equal to the
direct method. And for eflBciencics higher than 26.4 per cent,
the warmmg machine is thermodynamicaUy superior.
CHAPTER XVIII
STEAM TURBINES
194. The detailed descriptions of the various types of steam
turbines and the attendant mathematical discussions require an
extended treatise. For such a treatise the reader is referred to
Dr. A. Stodola's classical work, *' Die Dampf-turbinen." * No
attempt will here be made to do anything further than lay down
the most elementary principles, so as to enable the student to
take up reading matter, on the subject, of an advanced nature.
In steam turbines, as well as in water tiu^bines, there are impulse
turbines and reaction turbines. However, in the case of water
wheels, the types most frequently used are single stage; i.e., one
stationary part, which carries the guides, by means of which the
water is given the proper direction before entering the wheel, and
one rotating part, carrying a number of curved blades, by means
of which the energy stored in the water due to pressure and velocity,
is absorbed. On the other hand steam turbines must be multi-
stage, i.e., consist of a number of fixed parts called guides, and a
nimiber of rotating wheels, called rotors; otherwise the speed
would be impracticably high. Fig. 57 is a diagranmiatic repre-
sentation for two stages of a multi-stage turbine.
196. Impact on Curved Surfaces. Before proceeding to make
any mathematical discussions for steam turbines, it will be well
to consider a few cases for a non-expansive fluid, such as water,
impinging on curved surfaces. Let, in Fig. 58, abc represent the
section of a curved blade, having impinging upon it a stream of
water with a velocity, relative to the earth's surface, represented
* Translated by L. C. Loewenstein.
289
290
THERMODYNAMICS
in magnitude and direction by ka. This velocity is briefly
designated as absolute velocity. The line ad represents in mag-
nitude and direction the velocity of the blade. Now, while a
GUIDE
J J J J
Fig. 57.
particle of water starting from a moves to e, a distance equal to
Fi, the tip of the blade a has suffered a displacement Vi, as repre-
sented by ad. Hence the velocity of the water Yt^ relative to
Fig. 58.
the blade, is given by (fe. Therefore, the absolute velocity at
entrance is equal to the vector sum of the velocity of the tip of the
blade, at entrance, and the relative velocity.
STEAM TURBINES 291
If the water is to glide onto the blade, so that there shall be
no shock, the tangent, to the tip of the blade at a, must be parallel,
as shown by ka, to the relative velocity Vr. If, now, there is
experienced no friction by the water as it glides along the blade,
and there is no sudden change in direction, then the magnitude
of Vr will not change; and the water will leave the blade, parallel
to the tip at exit, as shown by ci. If V2 is the velocity of the tip
at exit, then the vector sum of Vr and V2, gives for the absolute
velocity at exit, F2, as shown by cj. We then have, respectively,
for the triangles of velocities at entrance and exit, ade and dj.
If, from a we draw ag equal and parallel to V2, and close the tri-
angle by eg, we find Vt, the vector difference between Vi and V2;
i.e., the total change of absolute velocity. Resolving Vt into two
components, one normal to the motion of the blade and the other
parallel to the motion, we find Ve, represented by ef, the total
change of absolute velocity in the direction of motion. Ve may
be called the effective component, since this is the one producing
the motion. From the diagram, it is obvious that the absolute
velocity at entrance cannot be parallel to the motion of the blade,
but must make some angle with it; otherwise entrance into the
channel, included between abc and a'b'c', cannot take place.
Similarly, at exit, there must be a normal component to carry the
water away, so as not to interfere with the following blade. Fur-
thermore, it is essential that the two surfaces of the blade tips,
both at entrance and exit, come to a point and have, practically,
a common tangent, parallel to the relative velocity, so that the
following blade, a'b'c', may glide into the stream without shock.
Unless this is so, there will be a loss in efficiency; for, whenever
a stream of water impinges upon a surface with shock, there are
developed eddy currents which, when subsiding, develop heat and
the energy thus consiraied is dissipated to the surroundings.
To find the theoretical efficiency of a system of blades, as
depicted in Fig. 58, it is only necessary to take the ratio of change
in kinetic energy, in passing through the channel, to the kinetic
292 THERMODYNAMICS
energy at entrance. Let M be the mass of water, per unit time,
passing through the channel; then^the kinetic energy, at entrance,
is
TFi = ^ (1)
The kinetic energy at exit is
W..«^; (2)
and the energy given up to the sjrstem is
Wi-W2=W3=^(yi^-V^) (3)
Hence, the efficiency is given by
Wi Vr'-Vi^ ...
-"^Wz — F7~ ^^^
Equation (4) shows that V2 should be as small as possible; which
means that it must be normal to the direction of motion of the
blades and just sufficient to carry the required quantity of water
away from the channel.
We may consider this in another manner. Since force is
numerically equal to rate of change of momentum, the eflfective
force, in producing motion, is
F=MV,] (5)
where F* is the change of absolute velocity in the direction of
motion. And the power developed, since power is numerically
equal to the product of force and speed, is
P2^MVeVi) (6)
where P2 is the power developed. Equation (6) assumes vi and
V2 nimierically equal. The power of the stream before imping-
ing, since M is the mass of water which passes through the channel
per unit time, is
Pi=-^-; (7)
STEAM TURBINES
2d3
where Pi is the power delivered by the stream. Taldng the ratio
of Pa to Pi, we find for the eflSciency,
10 -pj- Vi^
(8)
Equation (8) again shows, that if Vi and vi are fixed, the efficiency
is a maximum when F« is a marimmn; i.e., the velocity com-
ponent normal to the direction of motion is as small as possible.
196. The Pelton Cup. Pelton wheels may be taken as repre-
senting impulse tiurbines in the case of hydraulic motors. One
of the cups, as used in this tjrpe of wheel, is represented diagram-
matically in Fig. 59. Vi is the absolute velocity of the entering
Fig. 59.
jet, and vi the velocity of the cup, whose section is represented by
abc. The velocity of the water relative to the cup is given by
Vr^Vl-Vl.
(9)
Since the direction of motion of the water at exit makes an angle
6 with the direction of motion of the cup, the component of the
velocity of the stream at exit, parallel to the motion of the cup,
is given by
7,= (Fi-t;i) cos e (10)
294 THERMODYNAMICS
And the absolute velocity, parallel to the direction of motion of
the cup, at exit, is
F2'=t;i~(7i-t;i)cos e (11)
The change in absolute velocity in the direction of motion, there-
fore, is
Fa = 7i-72' = Fi- {vi-(Fi-t;i) cos 61
= (7i-t;i)(l+cose). . . (12)
If M is the mass of water, per unit of time, impinging on the cup,
then, the force moving the cup is nmnerically equal to the product
of change in velocity and mass; hence we find for the force acting,
F=M(Fi-t;i)(l+cose) (13)
Finally, since power is numerically equal to product of force and
speed, we have, by multiplying both sides of equation (13) by vi,
for the power developed by the cup,
P2^Fvi^Mvi{Vi-vi)(l+Q0S%) (14)
It is obvious, from Fig. 59, that the motion of the water cannot
be completely reversed; i.e., the direction of motion of the water
leaving the cup must be inclined to the direction of motion of the
cup. For, otherwise, the stream at exit will interfere with the
forward motion of the following cup.
By assmning, in equation (14), the power and the velocity of
the cup variable, and the other quantities constant, we find, by
differentiating for a maximum,
^=M(Fi~2t;)(l+cos e) = 0;
(W
from which
That is, for maximum power, the velocity of the cup must be one-
half the velocity of the stream.
STEAM TURBINES 295
Substituting in equation (14), this value for vi, we find, for the
maximum power developed by the cup,
P2'-M^(vi-^yi+cosii)=^^{l+<ios%). . (15)
Since the energy of the impinging jet is
Pi = —2-'
the efficiency of the cup becomes,
- ^2 ^ Fi''(l+cos 6) _ 1+cos e
Pi 271=
"IQ D OT7 2 O .... m)^
We may deduce the expression for the eflSciency of the cup by
considering the kinetic energy at entrance and exit. By an inspec-
tion of Fig. 59, it is obvious that the absolute velocity at exit is
the vector sum of (Fi— «i) and vi. Designating the absolute
velocity at exit by F2, we find
722 = (7i-t;i)2+t;i2_2(Fi-t;i)t;i cos 6
= Fi2-2(Fit;i-vi2)(l+cos 6).
And the eflSciency becomes, since it is given by the ratio of the
kinetic energy absorbed by the cup, to the kinetic energy of the
stream,
__ 2( Fit;i-t;i2)(l+cose)
Assmning Vi and 6 constant, and the velocity of the cup and the
efficiency variable, the expression for the efficiency becomes,
_ 2(yit;-y2)(l+cose)
and solving this for a maximimi, we find
dT3_ 2(Fi-2t;)(l+co8 6) _
from which
Vi
^=2-
m lis RS^Skamics
'tS|'ifj|^8f:W'^»^|gyiuation for efficiency, we find
Bi8)ttJ9v7llf|1RCsVigl^ ^ equals zero, so that the
AjflB iQtSjK^A^jljij^^versed, the efiSciency equals
j8li.^tfi'Wiwrfl|i^',^^necessarily eo, since for max-
M'^Si'l llJl^'w/'SfH^^^i' i^ brought to rest; and
lg^H^il^J5E£^8^^_G||9, before impact, is completely
flr^tM .gkiWJBiiM^yii as previously stated, it is
to zero, on account of inter-
.^■Vssume the angle 6 to vary
id 90°. Then, as just stated,
_ ;rf«||iiity; andfortheanglee=90'',
^^?S*^«^3 equals 90°, the result is equiv-
-"^^ t^ ^E>ck, as represented in Fig. 60.
•^■^^s impact, then, since for maxi-
^sup equals Vi/2, the absolute
i^^^^^Dpifttifrifp^V is equal to Fi/V^. There-
STEAM TURBINES 297
fore, since the efficiency is equal to the ratio of ihe kinetic energy
absorbed by the cup to the kinetic energy of the stream, before
impact, we find
^- 7i2 2 ^^^^
Hence, theoretically, the efficiency of an impulse turbine may vary
between 50 and 100 per cent; depending upon the value of the
angle 6. In practice it is attempted to have the angle 6 just
sufficiently large so that the stream, leaving the cup, does not
interfere with the following cup. In properly designed impulse
wheels the actual efficiency may be, considering all losses, as high
as 90 per cent. The results deduced clearly indicate, that in any
case, it is essential, if a high efficiency is to be realized, to reduce
the absolute velocity of the impinging stream, in going through
the turbine, as nearly as ppssible, to zero. And this, in the case
of steam turbines, is just as necessary a prerequisite for high
efficiencies, as it is in the case of water turbines.
The question, why is it possible, in the case of hydraulic
motors, to convert so large a fractional part of the theoretical
energy, due to the difference in topographical level, into actual
work, and in the case of heat motors, so small a fractional part
of the energy, due to the difference in " temperature level,'*
naturally suggests itself. The answer is obvious. Every heat
motor must act periodically. Even though the identical working
substance is not used in the succeeding cycle, the result is just the
same. For, the condition of the working substance, for the best
results, must be at the beginning of each cycle the same as it was
at the end of the preceding cycle. This is equivalent to cyclic
operation. In the case of the hydraulic motor, however, the
cycle is only partially completed. That is, the water, after
having performed work, in falling through a certain height, is
restored to its original condition by the action of the sun, which
completes the cycle automatically. In other words, the water
298
THERMODYNAMICS
at the height Hi, of the headrace, falls to the height H2, of the
tailrace, and performs, theoretically, the amount of work
w{Hi-H2);
where Hi is the height of the headrace, H2 the height of the tail-
race, and w the weight of water. But, to complete the cycle, the
water must again be raised from the level H2 to Hi. This is
done at the expense of the radiant heat from the sun, by means of
which the water from streams, lakes, the oceans, etc., is evapo-
rated and carried, by means of convection currents, to higher
elevations, where condensation takes place, and the difference in
elevation, Hi'-H2, is again established.
197. Flow of Fluids in Pipes of Varying Section (De Laval
Nozzle). The flow of a gas or vapor, under steady conditions,
Fig. 61.
through a pipe of varying cross-section, is very similar to the
flow of a liquid under similar conditions; but, the flow of a gas
differs materially from that of a non-compressible liquid in two
respects. That is, in general, the weight of the gas, or statical
head, is negligibly small in comparison with the pressure head
and velocity head; but, on the other hand, account must be taken
of the expansion.
Consider a pipe CC, such as is represented in Fig. 61, and
assume a steady flow, i.e., the mass of gas entering the section at
C, for a given interval of time, is equal to the mass leaving the
section at C, during the same interval of time. And further,
that the gas flows, without friction, in straight stream lines.
Let m be the mass of gas that enters the channel at C, per unit
STEAM TURBINES 299
time, with a speed Sa, and pressure pa, then an equal mass will
leave the channel at C, during the same interval of time, with
' some speed St, and pressure p^. Assiune the pipe CC, to be divided
into an indefinitely large number of sections, such that the thick-
ness of each section is indefinitely small. Let the pressures on
the left-hand side of the various sections, be, respectively,
P«> P2, pa, . . . Pn-l, Pn]
and on the right-hand side of the sections,
P2, ps, P4, . . . Pn, Pb.
Representing the respective cross-sectional areas by
Ao} -^.2) -A3, . . . A-n) -Aft,
and the corresponding thicknesses of the elements by
dsa, ds2f dssy . . . dsn,
then the work done by the positive pressiu-es, during the time that
the displacement dsa takes place at C, and the displacement dsn
takes place at C, is
W = PaAad8a+P2A2d82+ ' ' ' +Pn^lAn-ldSn-l+PnAndSn. . (18)
Likewise, the work done by the back pressures is
W''=—p2A2ds2'-p^A^dsz—'''—pnAndsn—pbA^4sb^ . . (19)
Taking the sum, of equations (18) and (19), we find, for the net
work done, due to change in pressure,
Wi = W'+W=PaAadsa-pi>Ai4si>. . . . (20)
Changing, in equation (20), the subscripts a and 6 to 1 and 2,
we have
Wi=piAidsi'-p2A2d82 (21)
300 THERMODYNAMICS
If dsi is the distance passed through^ at C, in the time dt^ and
ds2 the distance passed through at C in the same interval of time,
then
dsi = sidty
and
ds2=S2dt;
where si and 82 are, respectively, the initial and final speeds.
Substituting, in equation (21), these values of dsi and d82, we obtain
Wi = (piAi8i-p2A282)dt (22)
Since, m is the mass of gas, flowing per unit time, we have, for
steady flow,
A181 A 282
where vi and V2 are, respectively, the volumes per unit mass of
the gas corresponding to the pressures pi and p2. Substituting
in equation (22), we obtain
Wi = {viVi'-p2V2)mdt (23)
The change in kinetic energy is
and expressed in engineer's units, this becomes
W2 = ^-^^mdt (24)
The work due to expansion is,
Wz = (mdt)]vdv] (25)
and for the assumed conditions,
Wi+W2+Wz-=0.
STEAM TURBINES 301
Hence,
S^ So C*^
(piwi —p2V2)mdt-\ H — mdt+ (rndt) I pdv=0;
from which
— 2^ — =PiVi-p2V2+j pdv (26)
But,
rrPi
pdv= I vdp;
hence,
82^ -Si^ f Pi , .^^.
-"^^=Jp/^P (27)
If, now, the process be such that the equation
pv^=k
L IL
holds, then v=^k^p »; and by substitution, equation (27)
becomes
52^-
— = fcnj p ridp = —^kn{pi n -pg n ). . (28)
2g
1_
Substituting for k^ its value, we find
82^ — 81^ n L !L=i !Lii. n
Or, since
L L L
i_
equation (28) may be simplified by dividing by i» , and multiply-
ing the first term in the parenthesis by pi» vi, and the second term
by ^2" f2. Performing this operation we find
—2^ ;^(Pifi-P2r2) (30)
302 THERMODYNAMICS
If the pipe is curved, then the pressure on the convex surface
is less than on the concave surface. However, unless the change
in direction is considerable, the difference in pressure is very small.
The foregoing conclusions are the result of a modification of
of Bemouilli's theorem; i.e., applying the theorem to a compress-
ible fluid of negligible weight.
If the initial speed is negUgible, as is the case when discharge
takes place from a comparatively large vessel, then si, in equation
(29), may be omitted, and we have
n-l
82^
2g
Solving for 82, we find
-„-V.{i-(i!) } (31)
Now, the mass of fluid conveyed, per unit time, through any
section is
As
m= — :
V
where A is the area of the section, s the speed, and v the volume
per unit mass. But for steady flow this is a constant throughout
the pipe. Hence we have
^^As^A^2 (3
V V2
where A 2 is the area corresponding to the pressure p2, and V2
the corresponding volume per unit mass. By combining equa-
tions (32) and (33) we find
n-l 1
A2S2_A2\ 2gn
m
V2 V2
[„^.''-{-(i:) " W- <3«
Assuming again, the flow to be such that the equation
pi>"=pi»i" (35)
STEAJVI TURBINES 303
holds, we find
Substituting this value of V2 in equation (34), we find
Since, as previously stated, m is constant, equation (34) may
be written
2 n+l 1
m
-t-4»^F:{©-'-C-.)"}f^ <="
where A is any section, p the corresponding pressure, and v the
corresponding volmne per unit mass. Since As/v is a constant,
s/v must be a maximum, when A is a minimum. But, s/v becomes
a maximum, and hence, A a minimum, when the value of the
expression included in the bracket, of equation (37), becomes a
maximmn. Hence, we may write
and
2-n 1_
^dx_2 p n n+l p^ _^
Pin pi n
from which
P
M^+iY'^ (38>
The value of p, as given by equation (38), is that value which
makes s/v a maximmn, and, therefore, A a minimum. Substitut-
ing this value of p, for p2, in equation (32), and reducing, we find
304 THERMODYNAMICS
Finally, by means of equations (35), (38), and (39), we find
^=^«=d(i5Lp,„,)LA(^pi(4-y"^F. (40)
V v\n+l I [n+lt;i\n+l/ J ^ ^
Equating the right-hand members of equations (36) and (40),
we find
4?
A
n+l\n+l/
kg)" -©'
(41)
which gives the ratio of the area, for the pressure p2, to minimum
area.
Solving equation (38), on the assumption that the fluid is
saturated steam, for which n equals 1.135, we find
p=0.577pi.
By means of equation (32), the final speed may be determined,
when the initial and final pressures are known; and with the aid
of equation (37) any one of the three quantities, viz, m, A, and
p, may be found, if two of them are given.
Since, in deducing the foregoing equations, we equated work,
expressed by the product of pressure and volume, against energy,
expressed by the product of mass and square of the speed, we must
in substituting numerical values, in these equations, use the same
system of units. As an example, if in equation (39), s is to be
given in feet per second, pi must be given in lbs. per square
foot, and vi in cubic feet. Reducing equation (39), so that pi
is expressed in lbs. per square inch, and substituting for g and
n the proper values, we find
/2X32.2X1.135^-,. \2
s= — -. ,^- Xl44pii;i I =70.2(piri)J.
\ Z.ioO /
STEAM TURBINES 305
For steam under a pressure of 100 lbs. per square inch, the volume
per pound is approximately 4.43 cu.ft.; hence, by substitution,
we find
5 = 70.2(443)i = 1478 ft. per sec.
If the pressure be 200 lbs. per square inch, for which the volume,
per poimd, is approximately 2.29 cu.ft., we find
8 = 70.2(458)1 = 1503 ft. per sec.
These two computations show that the variation in speed is small
when compared with the variation in pressure.
Reducing equation (40) in a similar manner, we find
2 1
A J 2X32.2X1.135 / 2 \o.i35 p,
^"144( 2.135 V2.135/ ^^ t;i
2
1
= 0.30A ( — j pounds per sec. ;
where A is now in square inches.
198. Two Principal Types of Turbines. From the previous
discussions on the flow of steam through pipes it is obvious thet
the speed of flow, for any considerable difference in pressure, is
very high; and that if any single-stage turbine, i.e., a turbine
consisting of a set of nozzles and only one rotating part, were to
«
utilize practically all the kinetic energy of the steam, due to its
speed at nozzle exit, the speed of the turbine would have to be
abnormally high. As an example, some of the De Laval tur-
bines, which were single stage, had speeds as high as 40,000 r.p.m.
Though the efiiciencies of the De Laval turbines, from the stand-
point of steam consumption, were not exceptionally low, the
enormously high speeds were a serious disadvantage. For, in
no other mechanical contrivance, not even dynamo electric
machines, which are operated at relatively high speeds in com-
parison with other machines, are such high speeds ever approached.
306 THERMODYNAMICS
Therefore, to utilize the power developed by a single-stage
turbine it is necessary to employ a reduction gear. But, a reduc-
tion gear means an additional first cost, and a lowering of mechan-
ical efl&ciency. Furthermore, proper lubrication becomes exceed-
ingly difficult when machines are operated under speeds such as
are attained by single-stage tm'bines.
The difficulties, however, stated in the preceding paragraph,
were overcome by the introduction of nudti-stage turbines.*
That is, by allowing the steam to act successively upon the rotors
of a multi-stage tiu-bine, its speed is gradually reduced, and the
speed of the turbine need not be abnormally high. The turbine
must of course be so designed that the steam expands, and the
temperature is reduced continuously to the lowest possible value
at exit. That is, the kinetic energy of the steam at exit must be,
as nearly as possible, equal to zero. ,
There is then the choice of the following types of multi-stage
turbines: Combined impulse and reaction, and impulse.
199. The Parsons Turbine. The Parsons turbine, at the pres-
ent time, represents one of the commercially successful types of
turbines; and may be considered a combined impulse and reaction
turbine of the parallel-flow type. That is, the steam passes through
the first set of guide blades approximately parallel to the shaft
of the turbine, and has given to it the proper direction so that
it may enter into the channels of the first rotor without shock.
At exit from the first rotor, the steam enters a second set of guide
blades, where it is again directed so as to properly enter the chan-
nels of the second rotor, etc. In this way the steam reacts,
expands, and falls in pressure continuously as it travels, from
* There appears to be considerable confusion in regard to the meaning of
the word '' stage." In some cases authors designate a turbine, as an n-6tage
turbine when there are n rotors, which is consistent with the nomenclature
employed in the case of hydraulic turbines. In other cases, however, namely
the Curtis turbine, by number of stages is meant the number of sets of
expanding nozzles.
STEAM TURBINES 307
entrance to exit, through the turbine. Since the steam is con-
tinually expanding, the length of the blades and spacing, for both
guides and rotors, must be increased so that the ratio of steam
speed and blade speed, upon which the efficiency of the tin-bine
depends, is maintained constant.
200. The Curtis Turbine. The Curtis turbine is of the impulse
type. The steam expands in a set of nozzles, where the pressure
head is converted into velocity head, and then impinges on the
ciUT^ed blades of a rotor. Part of the kinetic energy of the steam
is absorbed by the first rotor; the steam then at reduced speed
passes through a set of guide blades where it is directed so as to
properly enter the channels of a second rotor, where the speed
is still further reduced, etc., until, finally, the speed is very low.
The steam is then expanded through a second set of nozzles, and
again passes through a series .of rotors and guides, precisely as
in the first stage. This is continued until the pressure of the
steam has been reduced to the desired exhaust pressure. The
niunber of stages, other things being equal, depends, of course,
upon the range in pressure. Due to the fact that the speed of
the steam is reduced in each rotor, the passages traversed by the
steam must be continuously enlarged. This is brought about
by reducing the curvature of the blades as well as lengthening
them.*
20l. Comparison of Parsons and Curtis Turbines. Since
the speed of the steam entering a Parsons turbine is moderately
low, and for high efficiency its absolute velocity at exit must
approach zero in value, it follows that the relative velocity must be
high. That is, the relative velocity at exit being, approximately,
the vector difference between the absolute velocity at entrance
and the velocity of the wheel, it follows that, since the velocity
of the steam at entrance is low, the relative velocity at exit will
be high, and therefore, the velocity of the wheel must be high
* For a comprehensive discussion on the design and testing of turbines
see "The Marine St^am Turbine" by J. W. Sothern.
308 THERMODYNAMICS
in order that the absolute velocity at exit may be low. On the
other hand, in the case of an impulse turbine, the velocity of the
blade, for the best efficiency, is approximately one-half that of
the entering steam. Hence it is obvious that, other things being
equal, the Parsons turbine is inherently a higher speed prime
mover than is the Curtis turbine.
202. Turbines and Reciprocating Engines. No matter how
operated the steam turbine is inherently a high-speed prime mover;
and since power is proportional to the product of torque and angu-
lar velocity, it follows that for equal output, the steam turbine,
with its high rotative speed, will be of smaller dimensions than
a reciprocating engine. Furthermore, where rotative motion is
required, which is usually the case, the turbine needs no connect-
ing rod and crank, as does the reciprocating engine. Again,
where electric generators are direct connected, as in power plants,
high speeds, up to a certain point, are desirable. Since, the power
developed is equal to the product of e.m./. and current, high-speed
generators, for equal output, will have a lower first cost and occupy
less floor space than low-speed generators. Finally, the turbine
has the further mechanical advantage of having a uniform turn-
ing moment. On the other hand there are certain cases where
low speeds are essential either to successful operation or economy;
under such conditions the reciprocting engine is superior. As
an illustration of this we may consider present conditions in marine
engineering. As previously stated, the turbine, using high pres-
sure steam is, for high efficiencies, inherently a high-speed prime
mover; on the other hand the propeller of a ship, is, for high
efficiencies, inherently a low-speed mechanism. On passenger
liners, the increased rates, which passengers are ready to pay for a
reduction of time in transit, more than pay for the increased cost
of operation. However, on freight steamers, such is by no
means the case; and it appears that for such steamers the recipro-
cating engine combined 'with a low-pressure tm^bine, as regards
economy, is at least equal if not superior to the turbine.
STEAM TURBINES
309
Thermodynamically, the steam turbine is far superior to the
reciprocating engine. For, in the turbine there is no alternate
heating and cooling of the surfaces with which the steam comes
into intimate contact. In other words, in the case of a turbine,
very shortly after starting, steady conditions will prevail; and the
incoming steam, therefore, does not come into contact with sur-
faces which have been previously chilled by the low-temperature
exhaust steam. That is, the steam changes gradually in pressure
and temperature from admission to exhaust. And this means that
w/yy///////7/^?77^>^ ,
V
Fia. 62.
there is no condensation excepting that due to radiation. Hence,
in a turbine, condensation is largely eliminated in comparison
with a reciprocating engine; and herein lies one of the great
factors that makes the turbine thermodynamically superior to
the reciprocating engine. Another important factor is the fact
that in a turbine a good vacuum is utilized to much better advan-
tage. This is illustrated by Fig. 62. Let, in the figure, ABODE
be the indicator diagram of a reciprocating engine operating
between the pressures as indicated by the points A and E, Then
the net work done by the engine is measured by the area ABODE;
and the net work realized by means of a turbine, for the same
limits in pressure, is measured by the suresiABOFE. If, now, the
back pressure be reduced, from that as represented by the line
310 THERMODYNAMICS
ED, to that represented by the line HI, the net gain in work,
by means of a reciprocating engine, is measured by the area
EDIH, and that, in the case of a turbine, by the area EFGH,
That is, due to mechanical considerations, the length of stroke
of the reciprocating engine is fixed; and hence, full expansion
cannot be realized. But, in the turbine full expansion is realized
and the toe^ of the indicator diagram is utilized in doing useful
work. This gain in work, in the case of low-pressure turbines,
is quite appreciable.
203. •Turbine Tests. It has been found impossible, up to
the present time, to devise any method by means of which to
determine the indicated power of a tm-bine, in the same manner
that the indicated power of a reciprocating engine is determined.
There is, however, no difficulty experienced in determining the
output, or brake power. The output is determined in precisely
the same manner as described in Art 133; and the thermal effi-
ciency is determined as described in Art, 136. If it be desired
to determine the commercial efficiency, the method of procedure
is precisely the same as that described in Art, 137. The com-
parison frequently made is that between the actual output of the
turbine, and that which would have been obtained on a Rankine
cycle, as discussed in Art, 138.
204. Reciprocating Engines and Low-pressure Turbines. In
1906, H. G. Stott presented a paper* before the American Institute
of Electrical Engineers on " Power Plant Economics," which gave
a complete analysis of the various losses, from the coal bunkers
to the bus-bars, of the Interborough Power Plant, located at
Fifty-ninth Street and Eleventh Avenue, New York City. The
prime movers employed at that time were of the Manhattan
type compound Corliss engines; two engines being connected to
one generator of 7500 K.W. maximum capacity.
The following quotation is an extract from Mr. Stott's paper,
"Power Plant Economics," Transactions of the A.I.E.E., Vol. XXV.
STEAM TURBINES 311
which is one of the most complete and instructive analyses that
has ever been made of a power plant:
" Three years ago the steam-power plant for the generation
of electricity had apparently settled down to an almost uniform
arrangement of standard apparatus in which one power plant
diflFered from another only in details of construction of engines,
generators, and auxiliaries. As only about twenty years had then
elapsed since the first central station was put in operation on a
commercial basis, this uniformity of design seemed to indicate
that in the near future it would only be necessary to purchase a
standard set of power-plant drawings, and make the necessary
changes in size of units in order to have a station of the best type
known to the art.
" The internal combustion or gas engine had from time to
time been brought forward as a candidate for the position of
prime mover, with every prospect of improved economy in fuel
consumption; but with the exception of a few special instances it
was not looked upon with favor, as shown by the almost universal
use of the steam engine.
" After a long period of development a new factor in power-
plant design; namely, the steam turbine, was placed on the market
in commercial sizes. It is safe to say that during the last three
years no other piece of apparatus has had so stimulating an effect
upon the power plant. Its effect upon the entire plant has been
most beneficial, for it has revived the apparently moribund
superheater. This has now been so developed and improved that
superheat of 200° or 300° fahr. can be safely and economically
obtained. With the development of the superheater further
study of the problem of combustion has improved the efficiency
of the furnace; and this most important subject is apparently
susceptible of still further development.
" One other important result of the steam-turbine develop-
ment has been the development of condensing apparatus to such
a point of efficiency that a vacuum within one inch of the simul-
312 THERMODYNAMICS
taneous barometer reading can now be maintained without
difficulty.
*' Another change in the power plant has been the reversion
to high-speed generators, resulting in decreased cost of the gen-
erator and its f oimdations, as well as saving in floor space.
" Last but not least the steam turbine has put the recipro-
cating engine and the gas engine on the defensive and has actually
been unkind enough to throw out hints in regard to the applica-
tion of Dr. Osier's proposed methods to the treatment of older
apparatus.
" The reciprocating engine and internal combustion engine
have not been slow in accepting this challenge; they have
responded by showing so improved an economy (especially in
the gas engine) that the situation has become most interesting
to the power-plant designer. It is safe to say that the develop-
ments of the next ten years will show very marked improvement
in power plant efficiency.
" In regard to this development the author wishes to direct
attention to the basic fact that in power plants one should not
look merely for increased efficiency in the prime mover, but should
also investigate and analyze the entire plant from the coal to the
bus-bars: first, in regard to efficiency; secondly, in regard to the
effect of load-factor upon investment; and thirdly, the effect of
the first and second upon the total cost of producing the kilowatt-
hour, which is the ultimate test of the skill of the designer and
operator.
'* Efficiency.
" In Table 1 will be found a complete analysis of the losses
found in a year's operation of what is probably one of the most
efficient plants in existence to-day and, therefore, typical of the
present state of the art.
STEAM TURBINES
313
((
Table No. 1
ANALYSIS OF THE AVERAGE LOSSES IN THE CONVERSION
OF ONE POUND OF COAL INTO ELECTRICITY.
1. B.T.U. per pound of coal supplied . .
2. Loss in ashes
3. Loss to stack
4. Loss in boiler radiation and leakage .
5. Returned by feed-water heater
6. Returned by economizer
7. Loss in pipe radiation
8. Delivered to circulator
9. Delivered to feed-pump
*10. Loss in leakage and high-pressure drips
11. Delivered to small auxiliaries
12. Heating
13. Loss in engine friction
14. Electrical losses
15. Engine radiation losses
16. Rejected to condenser
17. To house auxiliaries
Delivered to bus-bar
B.T.U.
Per Cent.
B.T.U.
14150
100.0
340
3212
1131
441
3.1
960
6.8
28
223
»
203
152
51
31
111
36
28
8524
29
15551
109.9
14099
14099
99.6
1452
10.3
Per Cent.
2.4
22.7
8.0
0.2
1.6
1.4
1.1
0.4
0.2
0.8
0.3
0.2
60.1
0.2
99.6
ti
Discussion of Data in Table 1
" Item 1, Bi.u, per Pound of Coal Supplied, The thermal
value of the coal used is evidently of prime importance, as it affects
the cost efficiency of the entire plant. The method of purchasing
coal used in the plant from which this heat balance is derived is
that of paying for B.t.u. only, with suitable restrictions on the
maximum permissible amount of volatile matter, ash, and sulphur.
'' A small sample of coal is automatically taken from each
filling of the weighing hoppers, so that the final sample represents
a true average of a boat-load of coal. This final average sample
is then pulverized and tested for heat value in a bomb calorim-
eter, after which a proximate analysis is made of another por-
314 THERMODYNAMICS
tion of the sample. This method of purchasing coal has been in
use for two years, with highly satisfactory results.
" Item 2, Loss in Ashes. It is doubtful whether a further
saving in this item can be made, as the extra care and labor
necessary to accomplish any improvement would in all probability
oflFset the saving in coal.
" Item 3. Loss to Stack. This is one of the most vulnerable
points to attack, as the loss of 22.7 per cent, is very large. Recent
investigations show that promising results may be obtained by
the use of more scientific methods in the boiler room. In prac-
tically all cases iUwill be found that this loss is due almost entirely
to admitting too much air to the combustion chamber, result-
ing in cooling of the furnace. This result is usually produced by
'* holes " in the fire; these " holes '' may be due to several causes^
but usually are due to carelessness on the part of the fireman.
" Fortunately, a very valuable piece of apparatus has been
placed upon the market in the shape of a CO2 recording instru-
ment. The results of a series of tests made with this instrument
are shown in Figs. 63 to 66.
" Fig. 63 shows the average condition of a furnace using small
sizes of anthracite, with forced draught. The conditions are such
that approximately 40 per cent, of the thermal value is being lost.
" Fig. 64 shows what improvement may easily be obtained by
watching the CO2 record, and indicates a saving of about 19 per
cent, over the previous case.
'* In the combustion of the small sizes of anthracite it is neces-
sary to use a draught of not less than 1.5 in. of water; this breaks
the crust of the fire in the thin spots, allowing the air to come
through in such volumes that an enormous amount of heat is
wasted in raising the temperaturie of the surplus air and at the same
time causing inefficient combustion in the entire furnace.
" Fig. 65 shows a record taken from a stoker boiler whilst the
recorder was covered up to prevent the fireman from seeing the
record.
STEAM TURBINES
315
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PER CENT. COs RECORD
DEC. 18-19, 1905, BOILER NO. 29. AVERAGE PER CENT. CO, »4.7
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THERMODYNAMICS
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DEC. 19-20. 1905, BOILER NO. 29. AVERAGE PER CENT. COt =7.6
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STEAM TURBINES
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318 THERMODYNAMICS
" Fig. 66 shows a record taken from the same stoker boiler
with the fireman watching the CO2 indications, resulting in a sav-
ing of over 12 per cent. Later records show that even better
results than an average of 11.4 per cent, of CO2 can be
obtained.
" Fig. 67 shows the calculated losses in fuel corresponding
to various percentages of CO2 for three different temperatures of
flue gases.
'' From a consideration of the above tests it seems reasonable
to assume that the 22.7 per cent, loss to stack can, by scientific
methods in the fireroom, be reduced to about 12.7 per cent, and
possibly to 10 per cent.
" Before the installation of the CO2 recorder a long series of
evaporative tests was made to determine the most economical
draught to carry when a high-grade semi-bituminous coal was
burning on the automatic stokers. The results shown in Fig.
68 were so remarkable that they were repeated under different
conditions in order to confirm them. Since the installation of
the CO2 recorder, however, the explanation is apparent; as the
draught giving maximum evaporation per pound of combustible
corresponds to the point of maximum CO2, illustrating the inher-
ent difficulty of maintaining efficient conditions in the combus-
tion chamber with high draught. This is well illustrated by Fig.
69, showing the draught, per cent, of rating, and percentage
of CO2.
" Item 4' The loss in boiler radiation and leakage, amounting
to 8 per cent., is largely due to the inefficient boiler setting of
brick which, besides permitting radiation, admits a large amount
of air by infiltration. This infiltration will increase with the
draught, thus tending to exaggerate the maximum and minimum
points on Fig. 68. The remedy for this radiation and infiltration
loss is evidently to use new methods of boiler setting, such as an
iron plate air-tight case enclosing a carbonate of magnesia lining
outside the brickwork.
STEAM TURBINES
319
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DEC. 22-23, 1905-BOILER NO. 31 AVERAGE PER CENT C02«11.4
STOKER OPERATED
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THERMODYNAMICS
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CURVES SHOWING THE RELATIVE
TO PER CENT. CO« , BY VOLUME,
DIFFERENCE BETWEEN TEMPERA-
ENTERING AIR AND EXHAUST GAS
CURVE *'A"= 400" FAHf
1. "B"= 600 FAHI
M "C"= 600" FAHI
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STEAM TURBINES
321
" Mr. W. H. Patchell,* of London, who recently visited us,
has introduced very large boilers, assembling two in one setting;
each boiler has a normal evaporation of 33,000 lb. per hour
and in this way has cut down to a minimum the radiating surface
per square foot of heating surface. He has also introduced the
iron case with magnesia lining, and with good results.
" The question of boiler leakage is one in which the choice
of the lesser of two evils is necessary; for in the tubular or cylindri-
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CURVES SHOWING RELATIVE
ECONOMIES ATOIFFBRENT DRAUGHT IW
THE COMBUSTION OF SEMI-BITUMINOUS
COAL UNDEA A BABCOCK AND WILCOX
BOILER.UPON A RONKY STOKER.
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cal boiler the leakage will imdoubtedly be less than in the water-
tube type, owing to the smaller number of joints in the water
space. But these two advantages are offset by the increased
difficulty of construction, and the danger of using large boilers of
the tubular type, especially with high-pressure steam.
" It is now generally admitted that there can be no more
difference in the efficiency of different types of boilers under
* See paper read December 7, 1905, before the Institution of Electrical
Engineers, by W. H. Patchell.
322
THERMODYNAMICS
similar conditions than there can be in electric heaters, press
agents to the contrary notwithstanding.
" Item 5. Returned by Feed-water Heater. ^ The importance of
getting the feed water to the maximum temperature obtainable
is generally recognized, and would seem to indicate that all auxili-
HOUR - 23-HOUR RUN, JAN. t-2-06
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GRAPHICAL LOG OF AVERAGE READINGS
TAKEN ON BOILER NO. 31, JAN. 1, 2, 1906.
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aries should be steam driven so that their exhaust may be utilized
in the feed-water heater; in this way the auxiliaries may operate
at about 80 per cent, thermal efficiency.
*' Item 6, Owing to the difficulty of pumping water at tem-
peratures above 150 degrees fahr., when under pressure, it
becomes necessary to install economizers for the purpose of
increasing the feed-water temperature to 200 or 250 degrees fahr.
STEAM TURBINES 323
As this increase of temperature is obtained from the waste gases
at no expense for fuel, it only becomes necessary to consider the
load-factor, as will be shown later, in order to decide whether
economizers should be installed or not. In practically all cases
where the load factor exceeds 25 per cent, the investment will be
justified.
" In deciding upon the size of economizer to be installed it is
important to consider first, the influence of the economizer upon the
available draught due to the obstruction it oflFers and also due to
the reduced stack temperature; the second important consider-
ation is to equate the interest and depreciation charges against
the saving in fuel, and so determine the amount of investment
justified in each particular case.
" Item 7, Loss in Pipe Radiation, By the use of two-layer
pipe covering, each layer being approximately 1.5-in. thick, and
sections put on in such manner that all joints are broken, the
radiation losses have become practically negligible.
" Items 8 and 9. Heat Delivered to Circulating and Bailer-Feed
Pumps. As these auxiliaries may be either electrically driven or
steam driven it is interesting to note that the thermal efficiency
of the electrically-driven pumps would be equal to the thermal
efficiency of the plant, multiplied by both the efficiency of con-
version from the alternating to direct current and by the motor
efficiency. In this case, there would be a net thermal efficiency
of 10.3X0.93X0.90=8.63 per cent., whereas the thermal efficiency
of the steam-driven auxiliary discharging its exhaust into a feed-
water heater at atmospheric pressure would be approximately
87 per cent.
" Item 10, Loss in Leakage and High-Pressure Drips, The
loss in leakage should be infinitesimal, and the high-pressure
drips can be returned to the boilers, so that practically all the loss
under this heading is recoverable.
^^ Items 11 y 12 y and 17 are probably unavoidable and of so
small a magnitude as not to merit much consideration.
324 THERMODYNAMICS
" Item IS. Loss in Engine Friction, Recent tests of a 7500-
h.p. reciprocating engine show a mechanical efficiency of 93.65
per cent, at full load, or an engine friction of 6.35 per cent. As
this forms only 0.8 per cent, of the total thermal losses it is relatively
unimportant. Attention is called to the method of lubricating
all the principal bearings by what is known as the flushing system,
whereby a large quantity of oil is put through all the bearings
by gravity feed from elevated oil reservoirs common to all the
imits; after passing through the bearings the oil is returned by
gravity to oil filters in the basement and then pumped up to the
reservoir tanks again. About 200 gallons per hour are put through
each engine, and of this quantity only about 0.5 per cent, is lost.
This method of oiling undoubtedly contributes to the general
results.
" Item 14- As large electrical generators can now be obtained
which give from 98 to 98.5 per cent, efficiency, it would seem as
if the limit in design had been reached and that hereafter the
' problem of design is to be merely one of altering dimensions to
suit varying sizes and speeds. While this is true as far as the
efficiency is concerned, other problems are continually arising,
such as the design of generators for an overload capacity of 100
per cent, to meet the demand for apparatus capable of taking care
of great overloads economically for short periods, corresponding
to peak loads of a railroad or lighting plant.
*^ Item IS. Engine Radiation Losses. This source of loss has
evidently been reduced to a negligible quantity by the use of
improved material and methods of heat insulation.
" Item 16. Rejected to CondenseTf 60.1 per cent. This imme-
diately introduces the thermodynamics of the steam engine, a
subject so broad that it will be impossible to do more than touch
upon some of the most important points in considering steami-
engine efficiency.
" The efficiency * of any heat engine can be expressed by the
• Defined as ideal coeflficient of conversion, Art. 89. — ^Author.
STEAM TUBBINES
325
ratio of ^=— i~ — 2. where Ti is the absolute temperature of
ii
the steam entering the engine and T2 the absolute temperature
of the steam leaving the engine. Thus in the engine whose
steam-eonsmnption curve is given in Fig. 70, if the initial pressure
is 175 lb. gauge and the vacuimi at the low-pressure exhaust
837-560
nozzle is 28 in., then the maximum thermal efficiency is
837
=33 per cent. This would be true for any form of engine or
turbine working between the same temperature limits.
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A. ECONOMY CURVE FOR 7 500 H.P. ENGINE LOAD
EQUALLY DIVIDEO BETWEEN CYLINDERS.
B. ECONOMY CURVE POR 7 600 H.P. ENGINE LOAD
UNEQUALLY DIVIDED BETWEEN CYLINDERS,
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LOAD: KILOWATT- HOUP ( SWITCHBOARD READING)
FiQ. 70.
7 000
" In Pig. 70, however, it is seen that the point of maximum
economy shows a steam consumption of approximately 17 lb.
per kilowatt-hour, which is equivalent to 20,349 B.t.u. per
hour. One kilowatt-hour is equal to 3412 B.t.u. per hour,
so that the actual efficiency of the steam engine and generator
ig-Jil?. = 16.7 per cent. As the generator efficiency at this
20349 ^
326 THERMODYNAMICS
load is approximately 98 per cent, the net engine thennody*
namic efficiency * is 7r^= 17 per cent.
" The difference between the theoretical efficiency and the
actual is then 33—17 = 16 per cent., of which 0.8 per cent, has
already been accounted for in engine friction, so that the balance
of 15.2 per cent, is due to cylinder condensation, incomplete
expansion, and radiation.
'' As the engine friction in a two-bearing engine with high-
pressure poppet valves and low-pressure Corliss valves has by care-
ful design been reduced to less than 0.8 per cent, gain cannot be
expected here, so attention must be centered on the loss due to
cylinder ccnJensation, etc., amounting to 15.2 per cent., in order
to effect any improvement.
'' Superheated steam is the only remedy at hand and with it
we can probably effect an improvement of 5 or 6 per cent, by using
such a degree of superheat in the boilers that dry steam will be
had at the point of cut-off in the low-pressure cylinder.
'' Any greater amoimt of superheat than this will merely
result in loss to the condenser; for it should be remembered that
the cylinder losses increase with the difference in temperature
between the steam and exhaust portions of the cycle; in other
words, the greater the thermal range of temperature the greater
the condensation loss. This would seem to point to the use of
more cylinders; but this involves additional first cost and fric-
tion as well as more space and higher maintenance charges.
" Fig. 71 shows what may be gained by reducing the temper-
ature at the end of the cycle by means of increased vacumn, but
in the case in point the maximiun vacuum obtainable in practice
was used so that no additional economy can be expected in this
way.
* Defined as thermal efficiency under Art, 136. — ^Authob.
.T Balance.
ig reciprocating engines
;i|r.g;oiiS^j|f-S-. 12%
ffl |i^jftJi&1Se2^S*ll^fce of superheat 6%
ni^f* •]•"■»" -'2"§*
ijji'.jjj. . j^. .j^. .^. .^. ^^: •*■
S^Cwi^t^^W^il^ efficiency of the entire
4|€n^;i^r|^e total thermal efficiency
i«!apacity of the plant waa
■^ take care of the increas-
Li^KcVraffic; and it was finally
■^I^£o operate on the exhaust
328 THEEM0DYNAMIC8
•
The following quotation * indicates the all-around gain by this
combination.
" During the year 1908 it became apparent that owing to the
ever-increasing traffic in the New York subway, it would be
necessary to have additional power available for the winter of
1909-1910.
" 2. The power plant of the Interborough Rapid Transit Com-
pany, which supplies the subway, is located on the block boimded
by 58th and_59th Streets, and by 11th and 12th Avenues, ad-
jacent to the North River; it contains nine 7500-kw. (maximum
rating) engine units, besides three 1250-kw. 60-cycle turbine units
which are used exclusively for lighting and signal purposes.
" 3. The 7500-kw. units consist of Manhattan-type com-
pound Corliss engines, having two 42-in. horizontal high-pressure
cylinders and two 86-in. vertical low-pressure cylinders. Each
horizontal high-pressure cylinder and vertical low-pressure
cylinder has its connecting rod attached to the same crank, so
that the unit becomes a four-cylinder 60-in. stroke compound
engine with an overhanging crank on each side of a 7500-kw.
maximum rating 11,000-volt, three-phase, 25-cycle generator.
The generator revolving field is built up of riveted steel plates of
sufficient weight to act as a flywheel for the two engines con-
nected to it. This arrangement gives a very compact two-bear-
ing unit. The valve gear on the high-pressure cylinders is of the
poppet type, and on the low-pressure of the Corliss double-ported
type.
'' 4. The condensing apparatus consists of barometric con-
densers, arranged so as to be directly attached to the low-pressure
exhaust nozzles, with the usual compound displacement circu-
lating pump and simple dry-vacuiun pump.
'' 5. These engine and generator units are in general probably
the most satisfactory large units ever built, as five years' experience
•"Testa of a 15,00(>-KW. Steam-Engine-Turbine Unit," by H. G.
Stott and R. J. S. Pigott. Transactionfi of. the A.S.M.E., VoL XXXII.
STEAM TURBINES 329
a
with them has proy^; their uormal economic rating is 5000
kw., but they operate equally well (water rate excepted) on
8000 kw, continuously,
'^ 6t In considering tjie problem of how to get an additional
supply of power, every available source was considered, but by
a process of elimination only two distinct plans were left in the
field.
" 7. The electric transmission of power from a hydraulic
plant was first considered, but owing to the high cost of a double
transmission line from the nearest available water power, and the
impossibility of gett ng reliable service (that is, service having
ft maximum total interruption of not more than ten minutes per
annum) from such a line, further consideration of this plan was
abandoned,
" 8. The gas engine, while oflFering the highest thermo-dynamic
eflBciency, at the same time required an investment of at least
35 per cent more than ordinary steam-turbine plant with a prob-
able maintenance and operation account of from four to ten times
that of the steam turbine,
" 9, The reciprocating-engine unit of the same type as those
already installed, was rejected in spite of its most satisfactory
performance, on account of the high first cost and small range of
economical operation. Reference to Fig, 72 will show that the
economic limits of operation are between 3300 kw. and 6300
kw,; beyond these .imits the water rate rises so rapidly as to
make operation undesirable under this condition, except for a
short period during peak loads,
" lOt The choice was thus narrowed dgwn to either the high-
pressure steam turbine or the low-pressure steam turbine. There
was sufficient space in the present building to accommodate three
7500^kw. units of the high-pressure type, or a low-pressure unit
of the same size on each of the nine engines, so that the
questions of real estate and building were eliminated from the
problem,
330
THERMODYNAMICS
"11. The first cost of a low-pressure turbine unit is slightly
lower than that of a high-pressure imit, due to the omission of the
high pressure stages and the hydraulic governing apparatus,
but the cost of the condensing apparatus would be the same in
both cases. The foundations and the steam piping in both cases
28 80
111
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16
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VARIATION OF WATER-RATE
& BEST RECEIVER PRES8URE
WITH LOAD .
ORIGINAL ENQINE
A:- WATER RATE
B:- REC. PR.
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8000 4000 5000
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Fig. 72.
6000
7000
would not differ greatly. The economic results, so far as the first
cost is concerned, would then be approximately the same, if
we consider the general case only; but in this particular instance
the installation of high-pressure turbines would have meant
a much greater investment for foimdations, flooring, switchboard
STEAM TURBINES 331
apparatus, steam piping and water tunnels, amounting to an
addition of not less than 25 per cent to the first cost.
'' 12. The general case of displacing reciprocating engines
and installing steam-turbine units in their place was also con-
sidered. The best type of high-pressure turbine plant has a
thermal eflBiciency approximately 10 per cent better than the best
reciprocating-engine plant, but the items of labor for operation
and for maintenance, together with the saving of about 85 per cent
of the water for boiler-feed purposes and the 10 per cent of coal,
reduce the relative operating and maintenance charges for the
steam-tiu-bine plant to 80 per cent, as compared to 100 per cent
for the reciprocating-engine plant.
** 13. Assuming that the reciprocating engine plant is a first-
class one and has been well maintained, about 20 per cent of its
original cost (for engines, generators and condensers) may be real-
ized on the old plant and so credited to the cost of the high-pres-
sure turbine plant. But on the other hand, if the high-pressure
turbine installation is to receive credit for the second-hand value
of the engines, it must also have a debit charge for 100 per cent
of the original reciprocatmg-engine plant which it displaced.
The relative investments, therefore, upon this basis would be
approximately equal for the high-pressure or the low-pressure
turbine; but 80 per cent of the cost of the original engine plant
would have to be charged against the high-pressure turbine plant,
as against an actual increase in value (to the owner) of the engine
by reason of its improved thermal efficiency, due to the addition
of the low-pressure turbine.
" 14. The preliminary calculations, based upon the manu-
facturers' guarantees for the low-pressure and high-pressure
turbines, showed that the combined engine-turbine unit would
give at least 8 per cent better efficiency than the high-pressure
turbine unit, so that it was finally decided to place an order
for one 7500-kw. (maximum rating) unit, as by this means we
would not only get an increase of 100 per cent in capacity, but
332 THERMODYNAMICS
at the same time give the engines a new l^ase of life by bring-
ing them up to a thermal efficiency higher than that attained by
any other type of steam plant.
" 16. The turbine installed is of the vertical three-^tage
impulse type having six fixed nozzles and six which can be operated
by hand, so as to control the back pressure on the engine, or the
division of load between engine and turbine. An emergency
overspeed governor, which trips a 40-in. butterfly valve on the
steam pipe connecting the separator and the turbine and at the
same time the 8-in. vacuum breaker on the condenser, is the only
form of governor used. The footstep bearing, carrying the weight
of the turbine and generator rotors, is of the usual design supplied
with oil under a pressure of 600 lb. per sq. in. with the usual
double system of supply and accumulator to regulate the pressure
and speed of the oil pumps.
" 16. The condenser contains approximately 25,000 sq.ft.
of cooling surface arranged in the double two-pass system of water
circulation with a 30-in. centrifugal circulating pump having a
maximum capacity of 30,000 gal. per hr. The dry vacuum
pimip is of the single-stage type, 12-in. and 29-in. by 24-in., fitted
with Corliss valves on the air cylinder. The whole condensing
plant is capable of maintaining a vacuum within 1.1 in. of the
barometer when condensing 150,000 lb. of steam per hr. when
supplied with circulating water at 70 deg. fahr.
" 17. The electric generator is of the three-phase induction
type, star-wound for 11,000 volts, 25 cycles and a speed of 750
r.p.m. The rotor is of the squirrel-cage type with bar winding
connecting into common bus-bar straps at each end. This type
of generator was chosen as being specially suited to the conditions
obtaining in the plant.
" 18. With nine imits operating in multiple, each one capable
of giving out 15,000 kw. for a short time, operating in multiple
with another plant of the same size, it is evident that it is quite
possible to concentrate 270,000 kw. on a short circuit. If we
STEAM TURBINES 333
proceed to add to this, synchronous turbine units of 7500-kw.
capacity, which, owing to their inherently better regulation and
enormous stored energy, are capable of giving out at least six
times their maximum rated capacity, the situation might soon
become dangerous to operate, as it would be impossible to design
switching apparatus which could successfully handle this amount
of energy. The induction generator, on the other hand, is entirely
dependent upon the synchronous apparatus for its excitation,
and in case of a short circuit on the bus-bars would automatically
lose its excitation by the fall in potential on the synchronous
apparatus.
" 19. The absence of fields leads to the simplest possible
switching apparatus, as the induction generator leads are tied
in solidly through knife switches, whichare never opened, to the
main generator leads. The switchboard operator has no control
whatever over the induction generator, and only knows it is
present by the increased output on the engine generator instru-
ments.
" 20. The method of starting is simplicity itself — the exciting
current is put on the engine generator before starting the engine,
and then the engine is started, brought up to speed and synchron-
ized in exactly the same way as before. While starting in this
way, the induction generator acts as a motor until sufficient
steam passes through the engine to carry the turbine above
synchronism, when it immediately becomes a generator and picks
up the load. Three of these 7500-kw. low-pressure turbine units
have been installed and tests run on Nos. 1 and 2. No. 3, having
been just started, has not yet been tested.
" 21. Instead of inserting in this paper the enormous accumu-
lation of data incident to these tests, we have divided the paper
into two parts in the hope that it would thus be more accessible
for reference, the first part giving the reasons for adopting this
particular type of apparatus, with a brief description of the plant
and a smnmary of the results obtained, and the second part con-
334 THERMODYNAMICS
taining all the principal data acquired during the tests, with
sufficient explanation to make their meaning clear without refer-
ence to the text."
" 24. The net results obtained by the installation of low-
pressure turbine units may be summarized as follows:
''a. An increase of 100 per cent in maximum capacity of plant.
" 6. An increase of 146 per cent in economic capacity of plant.
" c. A saving of approximately 85 per cent of the condensed
steam for return to the boilers.
" d. An average improvement in economy of 13 per cent over
the best high-pressure turbine results.
'' e. An average improvement in economy of 25 per cent
(between the limits of 7000 kw. and 15,000 kw.) over the
results obtained by the engine units alone.
"/.An average unit thermal efficiency between the limits of
6500 kw. and 15,500 kw. of 20.6 per cent."
206. Summary. The two preceding quotations are self-explan-
atory; hence no comment is necessary. But, before concluding,
it must be remarked that the internal combustion engine and the
steam turbine are still in the experimental stage; and that it is
impossible to predict what the final adjustment will be. * It is
true that the internal combustion engine has a higher thermal
efficiency than has any other heat motor. But, due to complexity
of construction, the internal combustion engine has a higher first
cost; and furthermore, its regulation is inherently inferior to a
reciprocating engine or turbine. Due to this, in spite of the fact
that the reciprocating engine has a lower thermal efficiency,
it still holds its place, on account of its simplicity and high over-load
capacity; the latter bemg especially important in most power
plants where it is necessary to take care of large "peak
loads."
It must be further remarked, that the installation of every
power plant is finally affected by the economy of transmission.
STEAM TURBINES 335
Whether power can be developed economically at any locality
depends upon whether the cost of power for the particular locality
is greater or less if developed at this particular point, or developed
at some other point and transmitted to the point under considera-
tion. This, of course, depends largely upon the economy of
transmission.
At the present time electrical engineers are giving consider-
able attention to the subject of high-tension transmission. And
if it develop that methods can be devised by means of which
corona losses can be eliminated, or partially avoided, for potential
differences far in excess of those employed at present, the subject
of power plant economics will require revision. For, if corona
losses can be avoided, the cost of power for any particular locality
will be materially changed. And hence, the cost for the produc-
tion of power will, likewise, be changed.
To illustrate concretely: Assimie that it becomes possible to
transmit with a potential difference of 300 kilo-volts instead
of 125 or 150 kilo-volts. Under these conditions the economy
of transmission is considerably increased; and the distances to
which coal can be transported, to compete with the increased
eflSciency of transmission, is considerably reduced. This, however,
is not the only governing factor. Ground rent also influences
the choice. That is, when the saving in transmission and the
saving in groimd rent, by locating the plant at the coal fields,
is balanced against the hauling of the coal, and the ground rent
for a large city, it may develop that it is more economical
to locate the power plant where the coal is mined. A similar
argument, of course, applies to water-power plants. That is,
the initial cost of a water-power plant is high, and therefore the
distance, for a given potential difference, over which power can
be profitably transmitted is limited; and, of course, the lower the
cost of transmission, the greater the area over which profitable
transmission may take place. Hence, as the potential difference,
which may be employed in transmission, is increased, the smaller.
336 THERMODYNAMICS
relatively, due to high ground rent, becomes the economy of a
localized plant. Therefore, if it develop, that potential differences,
far in excess of those employed at the present time, may be used,
power plants in large cities, where ground rent is high, will dis-
appear; and the future power plant will be located at the point
where the raw material, for the development of power, is found.
INDEX
PAOB
Absolute scale 43
Absolute zero 43
Adiabatic changes, change of dryness with 139
change of temperature with . .^ 59
Adiabatic equation for gases 50
Air, and adiabatic compression in compressor 235
and air compressors 230
composite diagram of compression, transmission, and expansion .... 260
compression and expansion 231
isothermal 233
constants of 230
loss of head in transmission pipes 254
Air compressor 230
throttling and other imperfections 249
Air motor, adiabatic expansion in 249
Air refrigerating machines 266
ideal coefficient of performance of 267
r-<|) diagram of 270
Air transmission system, theoretical efficiency of 262
Ammonia, coefficient of absorption 280
heat of dilution 281
Ammonia refrigerating machines, absorption 277
compression 271
Analysis of power plant losses 310
Anode 30
Boyle's law 41
departure from 43
Brake power 171
Brayton cycle 195
British thermal unit 9
Calorie 8
common 9
gram 9
mean 8
zero 8
337
338 INDEX
PAOB
Calorimeter 11
condensing 154
cooling constant of 14
Joly's differential steam 70
Joly's steam 69
separating 156
thermal capacity of 13
throttling 151
Calorimetry 11
Camot's cycle 97
a reversible process 100
coefficient of conversion 99, 109
steam operating on 116
with a perfect gas 104
Cathode 30
Centigrade scale 3
Characteristic equation 43
Chemical reactions of fuels 198
Clearance 156
equivalent length of ., 157
of air compressor 247
Commercial efficiency of steam engine 176
Compound engines, cross 189
tandem 183, 186, 188
Compressed air 230
Compressed air system, composite diagram of 260
theoretical efficiency of 262
Compression, adiabatic 235
isothermal 233
multi-stage 239
theoretical efficiency of 237
Condenser 121
Condensing calorimeter 154
Conduction 83
flow of heat along a bar 86
Conductivity, coefficient of 85
determination of coefficient of 86
in non-isotropic substances 89
non-homogeneous solids 89
of gases 90
of liquids 90
table of coefficients 90
Convection 83
Cooling, constant of calorimeter 14
Newton's law of 11
Stefan's law of 82
INDEX 339
PAGE
Critical temperatures, table of 63
Cross-compound engine 189
Curtis turbine 307
Curves, theoretical and actual 163
Cushion steam 157
Cycle 76
Brayton 195
Camot's 97
with a perfect gas as a working substance 104
Diesel 196
four phase 193
of perfect steam engine and boiler represented by means of T-<|>
diagram 137
Rankine's 177
reversible, as a standard 103
two phase 194
Cylinder feed 157
De Laval nozzle 298
Density and temperature of saturated steam 118
Dew point 66
Diesel cycle 196
Disgregation, heat of 62
Dissociation 28
Ebullition 61
Economics of power plants 310
and transmission 334
Efficiency and precompression 218
commercial 176
mechanical 175
of power plants and transmission 334
thermal 175
Elastic medium, propagation of wave motion 74
speed of 77
Elasticities, determination of ratio from speed of propagation of wave
motion 77
isothermal and adiabatic and ratio of two thermal capacities . 72
Electro-chemical equivalent 31
Electrolysis 29
counter e.m.f. of 31
Faraday's statement 30
Emissivity 81
Endothermic 21
Energy, principle of 33
340 INDEX
PAGI
Engine and boiler, perfect) with steam 121
studied with aid of the T—^
diagram 137
dearanoe 166
internal combustion 192
actual indicator diagram of 215
actual p-v and T— <J> diagrams of 222
double-acting cylinders 227
multi-cylinder 228
standard diagram 213
thermal efficiency of 214
!r-<|> diagram of 219
reversible 101
and refrigeration 110
as a standard 103
ideal coefficient of conversion 99, 109
simple thermodynamic 95
steam, condensing 150
double-acting 150
Elngines, compoimd 180
cross 189
double expansion 181
tandem, with large receiver 183
tandem, with small receiver 188
tandem, without receiver 186
triple expansion 190
heat 94
reciprocating and turbines 308
reciprocating, combined with low-pressure turbines 310, 328
Entropy 128
and temperature diagrams 134
change of 129
during reversible and irreversible processes 134
universal increment of 132
Evaporation 60
Exchanges, Provost's theory of 80
Exothermic 21
Expansion, adtabatic, in motor 249
change of drsmess during 167
double, in steam engine 181
external heat of 97
heat of 62
internal heat of 97
isothermal 233
triple, in steam engine 190
without doing external work 46
INDEX 341
PAGB
ExpanBion, linear 34
coefficient of 34
factor of 34
table of coefficients of 36
voluminal 35
determination of coefficient of, for liquids 36
direct measurement of 39
Fahrenheit scale 3
Four-phase cycle 193
Frequency 76
Friction brake 171
Fuel, calorific value of 203
determination of 204
Fuels, and chemical reactions 198
and fuel tests 198
liquid 207
table of calorific values of 208
Fusion, heat of 22
I
Gas, characteristic equation of 43
city 203
perfect or ideal 44
thermometer 44
Gases, adiabatic equation 50
change of temperature with adiabatic changes 59
conductivity of 90
elasticities of 72
expansion of, without doing external work 46
Gay-Lussac's law 41
general equations of 52
isothermal equation 41
table of constants 208
thermal capacities of 47
at constant pressure 48, 68
at constant volume 48, 69
determination of 68, 69, 70, 74
Gasoline 200
Gram calorie 9
Head, loss of, in transmission pipes 254
Heat, as a measiu'able quantity 7
content 125
effects of 33
engines or motors 94
flow along a bar 86
342 INDEX
Heati mechanical equivalent of « 17
from constants of air 48
of dilution 281
of disgregation 62
of fusion 22
of vaporization 23
for water ; 65
production of 33
total, of steam 65
unit quantity 8
Heating machine 287
Humidity, absolute 67
relative 66
Hydraulic-radius 255
Hygrometry 66
Impact on curved surfaces 289
Indicated power 173
Indicator 169
Indicator diagram 161
and valve adjustment 161
comparison of theoretical and actual curves 163
ideal of internal combustion engine 210
standard of internal combustion engine 213
Internal combustion engine 192
actual indicator diagram of 215
actual p^ and T— 4> diagrams 222
and double-acting cylinders 227
efficiency and precompression 218
multi-cylinder 226
standard diagram of 213
thermal efficiency of 214
Irreversible processes 27
Isothermal equation 41
Isotropic 35
Kerosene 201
Mean effective pressure 174
Mechanical efficiency of steam engine 175
Mechanical equivalent of heat 17
from constants of air 48
Mixtures, method of 10
Motor, adiabatic expansion of air in 249
reheating air before expanding in 251
Motors, heat 94
INDEX 343
PAOB
Multi-cylinder internal combustion engines 226
Multi-stage air compressors 239
Non-isotropic 36
Parsons turbine 306
Pelton cup 293
Power, brake ' 171, 214
indicated. 173, 214
Power plant economics 310
analysis of losses 310
efficiency and transmission^ 334
Precompression and efficiency 218
Pressure and temperature of saturated steam 119
Pressure, mean effective 174
Priming 126
Principle of energy 33
Radiation 80
Dulong and Petit's formula 82
Provost's theory of exchanges 80
Stefan's formula 82
Rankine's cycle 177
Reaumur scale 3
Reciprocating engines and turbines 308
engines exhausting to low-pressure turbines 310
Refrigerating machine 110
absorption 277
air 266
ideal coefficient of performance of 267
r-4) diagram of 270
commercial 266
commercial efficiency of 269
comparison of air and anunonia 284
compression, using volatile liquids 271
!r-4> diagram of 276
Kelvin heating 287
Refrigeration 265
and reversible engine 110
Refrigerator 94, 266
Resisted adiabatic expansion of steam 126
Reversible process, ideally 27
Reversible processes 25, 104
Separating calorimeter 156
Specific heat 9
344 INDEX
PAcn
Steam, behavior throughout the cycle 166
change of dryness during adiabatic expansion 126, 139
by means of T— 4»
diagram 139
cushion 157
cylinder feed 157
double expansion engine 181
dryness during expansion in cylinder 167
engines, reciprocating and turbines 308
engines, reciprocating with low-pressure turbines 328
exchange of heat with cylinder walls 168
jackets 170
operating on Camot's cycle 116
relation of temperature and entropy 139, 141
resisted adiabatic expansion 126
saturated, density and temperature of 118
relation or pressure and temperature 119
total heat of 65
triple expansion engine 190
turbine 289
unresisted adiabatic expansion 125
wire drawing 158
with perfect engine and boiler 121
work and incomplete expansion 145
work and superheating 148
work without expansion 144
zero curve 142
Zeuner's equation for adiabatic changes 151
Stefan's formula 82
Sublimation 23
Superheated vapors 66
Superheating 24
Tandem compound engines 183, 186, 188
Temperature 1
critical 62
table of 63
difference of 2
Temperature and density of satiu'ated steam 118
Temperature and pressure of saturated steam 119
Temperature-entropy diagrams 134, 219
dryness by means of 139
of air refrigerating machines 270
Temperatures, theoretical in cylinder of internal combustion engine 213
thermodynamic scale of 113
Thermal capacities, determination of by method of cooling 16
INDEX 345
PAGB
Thermal capacities, determination of by method of mixtures 15
Thermal capacities of gases 47
determination of, at constant pressure 68
at constant volume 69
ratio of, by method of Clement and De-
sormes 70
ratio of, from speed of propagation of wave
motion 77
Thermal capacity 8
by method of mixtures 10
per imit mass and specific heat 9
Thermal efficiency of internal combustion engine 214
of steam engine 176
Thermal equilibriimi 2
Thermo couple 6, 32
Thermodynamic drop 158
scale of temperatures 113
Thermodynamics, first principle of 93
second principle of 93
Thermometer 2
alcohol 6
gas 44
mercurial 5
resistance 6
Thermometric scales 3
Throttling and other imperfections of air compressor 249
calorimeter 151
Transmission of air and loss of head in pipes 254
of power and economy of plants 334
Turbine blades, impact on 289
Turbines, and reciprocating engines 308
comparison of Parsons and Curtis 307
Curtis 307
low pressure on exhaust of reciprocating engine 310, 328
Parsons 306
steam 289
tests 310
two principal types of 305
Two phase cycle 194
Unresisted adiabatic expansion of steam 125
Valve adjustment and indicator diagram 161
Vaporization 60
heat of 23
heat of, for water 65
346 INDEX
PAOB
Vapor pressures, addition of 61
Vapors 60
saturated 60
superheated 24, 66
Water equivalent 10
Water, relation of temperature and entropy 137, 141
Wave length 76
Wave motion, propagation of 74
speed of propagation 77
Wire drawing 158
Work, gain of, due to superheating steam 148
loss of, due to incomplete expansion of steam 145
loss of, due to using steam non-eacpaoaively 144
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